So a couple things. I think by your edit you mean 13th century? Genghis Khan didn't unite the Mongol tribes until then, the sack of Baghdad wasn't until 1258.

Ok, that being said, I think there is at least one thing wrong with the premise of your question which is in describing it as dogmatism vs. science, which I'll come back to.

The basic answer is that much of the ground work for this was already happening in what was called the Sunni Revival of the 11th-12th century, but the utter chaos that ensued from Mongol conquests basically created a situation where the more "dogmatic" theological frameworks could eventually win the argument. Sorry if this is long, I've done a good deal of reading into this and I think I can provide a pretty detailed explanation.

So to go into more detail: The pre-Mongol Situation. The Sunni Revival of the 11th-12th century is somewhat controversial; it's a theory, or a descriptive concept, not settled a fact or overt movement. What is settled fact is that in the 10th century virtually the entire Islamic world was under the political control of Shia dynasties, the Fatimids in the west, the Buyids in the east. The Buyids were succeeded by the Sunni Ghaznavids and the more famous Seljuqs.

But using "Sunni Islam" in this period is, itself, problematic. Sunni islam today is characterized by tolerant homogeneity, or "dogmatism" as you might call it. The four madhabs (schools of Islamic legal thought) co-existing in mutual respect. This concept did not exist in the 11th century. In the 11th century the conflict between the madhabs within Sunnism were often as tense and violent as those with the Shia. Much of this violence is taking place at a localized level as outside administrators seek to gain control of a populace by favoring one or the other competing school. One predecessor of Nizam al-Mulk (i'll come to him in a second) was executed after he tried to enforce the Hanafi madhab in Nishapur.

Now, Nizam al-Mulk (1018-1092) basically figured out a way around this problem with the creation of the Nizamiya, which were basically some of the world's first universities. Instead of forcing one school to dominate he just founded more schools and let them co-exist side by side. He even married one of his daughters to a Shia. This idea exploded across the Islamic world, such that you had dozens of Nizamiyya in major cities, and even in Fatimid Egypt, which within 200 years went from 0 to 73 of these madrasas i.e. it was a cultural phenomenon, not a political one.

So to take it back to your question, were these schools dogmatic? Anti-science? They were definitely religious schools. But they were also obviously educational institutions. They solidified the "dogma" of modern, homogeneous Sunnism, but through toleration. They did, however, produce the most famous "anti-science" muslim thinker, Al-Ghazali.

Al-Ghazali was basically the last major figure in a very large dispute between two groups, the Asharites, and the Mutazilites. One of the biggest problems was epistemological. The Mutazilites, taking after Greek philosophy, held that both the physical world as well as various extra-Quranic features of God could be examined and understood by human reason and logic. The Asharites disputed this, as it undermined the omnipotence of God. Ghazali's most famous work, The Incoherence of the Philosophers, accused the most famous Islamic Philosophers of heresy for their theological positions.

Today, Al-Ghazali is held up as the Arch-Nemesis of Reason, and Science, and as a promoter of rabid dogmatism by some people who don't know much about this history. Neil deGrasse Tyson gave a completely silly lecture on the subject. The problem is that it's reading history backwards. Al-Ghazali was expelled from his teaching position for pushing his Asharite theology. Islamic philosophers continued to publish, most famously Ibn Rushd's "The Incoherence of 'The Incoherence'" (which is also a fantastic title!) Also to say that the Mutazilites whom he was opposing were somehow heroes of science is a bit silly. Yes, they were rationalists, but their job was to reconcile philosophy and Islam. And Ghazali himself famously defended Sufism, an act of toleration. At the time of his death Al-Ghazali was hardly triumphant.

Phew, this brings us back to the Mongols. The mongols famously were not very Orthodox. A couple of the early Khans apparently tested every religion available to them. Their conversion to Islam seems to have taken place at the hands of the decidedly un-Orthodox Sufi mystics. Rather, their contribution was the complete chaos and utter calamity that they wrought in Muslim territories. The most prominent teachers in the Islamic world, the center of Islamic learning at the time, was in Persia. Here, the Mongols wiped entire cities off the map, most notably Nishapur which was utterly destroyed. The sack of Baghdad likewise resulted in huge losses of knowledge and culture.

As a result of this disaster two things happened. 1. You obviously had an enormous loss of life, where many of these scholars were simply killed, the books destroyed. 2. In the chaotic aftermath of the 14th and 15th centuries that's when figures like al-Ghazali have their triumph and Asharism simply "wins." The extent to which they "win" the argument is so great that there really aren't any serious theological disputes within Sunni Islam afterwards except maybe from Sufis, but the Orthodoxic debate is over, what's left is the Orthopraxic dispute.

So to sum up, no, the did not cause the shift ideologically in the sense that the debate was definitely already happening, its definitive figures were already there and had lived out their lives. But in the chaos afterwards the "dogmatic" view won out. Again, I put this in quotes because I think it's reading the question backwards, and you're talking about, basically, a dogmatic tolerance.

What they did cause, however, and which I view as being far more important in the end of the Islamic Golden Age, was simply the destruction of huge swaths of the Islamic world, including the most prominent intellectual centers. The modern forms of "fundamentalist" Islam look nothing like what existed around this time, even with the "dogmatic" philosophies. That doesn't come until movements like that of ibn abd al-Wahab in the 18th century. But again, even then, it's very difficult for me to look at the history of the Islamic world from the 15th-19th centuries and say that these were a dogmatic, scientifically backwards people. Literacy rates were apparently quite high, and the Ottoman empire kept pace with the European powers for centuries until realizing it was quite far behind in the early 19th, at which point it instituted a crash course of modernization. So in that sense I think the question is somewhat flawed as well.

Sorry, this has been extremely long and somewhat rambling, but I hope it helps. Some sources:

For the Sunni Revival Jonathan Berkey, The Formation of Islam, R. Stephen Humphries, Islamic History, A Framework for Inquiry, David Morgan, Medieval Persia 1040-1797 Islamic Civilization 950-1150 edited by D.H. Richards, particularly the sections by Richard Bulliet "The Political Religious History of Nishapur in the 11th Century," and George Makdisi "The Sunni Revival". Richard Bulliet's "Islam, a View from the Edge." and Daphna Ephrat's "The Seljuqs and the Public Sphere in the Period of Sunni Revivalism" which appears in The Seljuqs: Politics, Society, and Culture.

on the Asharites and Mutazilites: The Encylopedia of Islam is really useful here. As is George Hourani's "Reason and Tradition in Islamic Ethics."

On Mongol religion and conversion to Islam there are several, but I think “Sufis and Shamans: Some Remarks on the Islamization of the Mongols in the Ilkhanate" by Reuven Amitai-Press ought to cover the statements I've made here about Mongol conversion and religious practice.

edit: a word

Ok I'd like to take on some of what /u/CptBuck has said here. Mainly I'd like to argue that we actually see a flourishing of the sciences during the Mongol period, and that schools of Islamic though weren't irreparably destroyed by the Mongol conquest. I'd also take issue with his discussion of Sunnis and the rise of fundamentalist Islam undermining science, but other flairs are better qualified than me to do that.

Ok now onto the impact of the Mongol invasions on Islamic scholarship. We actually see science, and other scholarly pursuits, blossom under the Mongols. There are three areas I'm going to focus on cartography, medicine and astronomy.

Cartography: In the Ilkhanate the works of Rashid al-Din demonstrate an unprecedentedly detailed knowledge of East Asian geography including details like rivers, major cities, etc... An indication of its quality is the fact that it appears to have remained a standard work in the Islamic world for several centuries. It's worth noting that at the time scholars were aware the world was a globe, though this knowledge predated the Mongols. However while knowledge increased we don't see a change in the style of Muslim cartography, as they still kept their systems of longitude and latitude.

Medicine: Again the Mongols were keen on this as it had clear uses. With medicine while while we don't see the Islamic world adopting Chinese medical theory but certain practices were transmitted. Rashid al-Din published the first book on Chinese medicine in the Middle East and the Chinese practice of diagnosing by pulse spread quickly throughout the Islamic world. They also gained access to China’s knowledge of physiology, which was very advanced at the time. All this new knowledge was put to work, for example in the Houses of Healing in Tabriz, both eastern and western schools of medical thought were examined and practised.

Astronomy: In the Ilkhanate the study of the stars received a lot of official patronage. Hulagu, the first Ilkhan, built an entirely new observatory at Margarah, an undertaking which he took a personal interest in. The final building was well built with a library and a well equipped observatory. In this observatory Islamic and Chinese scholars worked side by side and shared knowledge. They came up with a giant compendium describing the positions and movement of the stars, the five known planets, the sun and the moon.

As Iran was an integral part of the Islamic world a large amount of these advances made their way to Islamic centres of learning. But those areas were doing just fine as well. By the time Baghdad was sacked it had been in long term decline and other centres of scholarship had arisen in the Islamic world e.g Cairo with the Al Azhar university. Thus anything lost in Baghdad was not lost to the whole Islamic world. I would continue to argue over his Sunni extremism causing the decline of Islamic science idea but as previously stated I'm hoping a better qualified flaired user can deal with that stuff.