The task of administering the late empire fell largely onto the shoulders of the literati class - the highly-educated, scholar-gentry men and families who provided the brain and man-power to staff an empire. In the Ming and Qing empires, these men of learning would generally compete in the various levels of the imperial civil service examination service, with those who successfully passed at the lower levels ultimately taking the test for the Jinshi degree in the metropolitan/palace exams. If they were successful in the highest levels of the exam, they could then be placed into various postings and positions within the empire. Individuals who did not pass at the highest level of the exam, or who had barely just scraped by in the Jinshi exams were seldom afforded official postings, but if they were, they were usually relegated to sidelined positions where they could hope for only modest rank and slow promotions. Unlike Europe, with vassal lords and so on and so forth, the Ming and Qing emperors had absolute control over their country, and the task of overseeing these lands were merely delegated to civil servants. The civil servants would do the emperor's bidding and in this way, the Chinese maintained a highly centralized government. It was indeed quite different from a feudal government.

So yes, China did indeed have an administration staffed by professional civil servants who theoretically were awarded postings and positions based solely on their meritorious service to the empire, as opposed to birthright or elected billets. China's civil administration was old, even by the time Zhu Yuanzhang proclaimed his era name of Hongwu and promulgated Ming rule over all of China. The civil service of the Song, Ming and Qing was designed to build a core of culturally-cohesive bureaucrats to man every nook and cranny of the massive Chinese imperial bureaucracy and unlike contemporary European governments, the way that this nucleus of officials was created was through comprehensive examinations and an emphasis on shared, common knowledge rooted in Confucian orthodoxy.

By the time of the Ming empire, there was... no real way that any aristocrat could really even hope to challenge the authority of the Imperial civil bureaucracy. That trend had been significantly diminishing since the Tang, and by the time of the Ming, members of the peerage largely only held nominal titles with a small allowance from the government. Outside of that, their only other roles were to just look pretty for ceremonial occasions. Their political powers were basically nonexistent. As the Cambridge History of China, Vol. 8 points out, the central Ming government really had a grip on power over China in a way that no preceding dynasty ever really had, and this hold on influence and power applies to the scholar bureaucracy too, especially when compared to the nobles outside of the royal family. Even imperial princes were largely powerless. Hongwu was a paranoid man, trusting few outside of his family and he enfeoffed many of his sons as vassal princes during his reign, including Zhu Di, who would usurp the Jianwen emperor to seize the throne as the 3rd Ming emperor in the mid-1402. But he was the exception. There would not be another successful princely rebellion during the Ming, and by the 1400s, princes no longer even had direct control of their own princedoms, instead merely living off of the stipends provided to them by the state and a portion of the revenues generated from their fiefs. All of the administering was done by the scholar-gentry bureaucrats, usually by prefects and provincial governors, with lesser officials serving under them. In England/Britain for instance, even into the 18th century, nobles and 'titled' persons were commonplace in the governmental structure, and in military matters, the British system of purchase of commission within the army stayed in place until Lord Cardwell's reforms in the late 19th century.

But to say that China's system was entirely meritocratic is unfair. By the mid-Ming, there was growing interest amongst families of means who were not of 'traditional' gentry-type families (the merchants) to send their sons off to pursue a career in the civil service. The purchasing of a position within an imperial university became increasingly common by the 16th century, and certainly by the time of the late Ming, people would outright purchase low-level degrees for their sons, in the hopes of providing them a head start to their examination careers and as an investment to bring honor to the family.

If you have access to resources through a university/library, Rui Wang produced The Chinese Imperial Examination System : An Annotated Bibliography. If you can find it, the Cambridge Histories also provide a very broad overview into the subject in their treatises on Ming and Qing government. Journals like Ming Studies also contain various academic essays on the matter, again if you have access to journal subscriptions. Here is an article that I briefly glanced over and it looks fine, though it focuses more on the exam system itself than the governance of China as a whole under civil bureaucrats.

To add briefly to /u/LateImperial's answer regarding the Ming, under the Qing the basic structure of the bureaucracy was the same, but there's a lot of extra caveats to account for.

Firstly, the elevated status of Manchus and dual administration of the Banner and Han populations added a little layer of complexity, as Bannermen were largely exempt from the jurisdiction of Han magistrates, while also making up a disproportionate amount of magistrates ruling over Han populations – as late as the final year of the Qing in 1911, Bannermen occupied 6% district magistracies despite making up 1% of the population, and were only more advantaged as you went up the scale, with half of the top 2000 jobs in the bureaucracy held by Manchus. While the proportion of Manchu provincial governors declined markedly after the Taiping war, their control over government was still disproportionately high.

Secondly, the civil-military line could and did become more blurred after the Taiping war. The war was largely won by Zeng Guofan's Hunan Army and to a lesser extent the Anhui Army and Chu Army of his protegés Li Hongzhang and Zuo Zongtang, respectively, but all three men had been civilian officials before the war began, and it was a marked break in Qing practice when Zeng was empowered to establish a consolidated militia force in Hunan in 1853. While Zeng disbanded his army, Li retained the Anhui Army and took it with him during his various subsequent employments, until it was effectively destroyed in 1894 during the Sino-Japanese War. Zuo Zongtang's army, bolstered by Hunan Army veterans, marched out to Gansu after the defeat of the Taiping to fight the rebelling Dungan Muslims, and then reconquered Xinjiang from the Turkic Muslim uprising, and with a bit of sabre-rattling resolved the Ili Crisis with Russia in the Qing's favour. Yuan Shikai pulled double duties as a civil mandarin and commander of the Beiyang Army in the years before 1911.

Thirdly, the Chinese-style civilian administration was not universal. Manchu administration – albeit long since reformed on Chinese lines – remained in the three Manchurian provinces of Fengtian (a.k.a. Liaodong), Heilongjiang and Jilin; a confusingly-named 'banner system' applied to non-Banner Mongols like the majority of the Khalkha, Khoshuts and Torghuts, who inhabited Mongolia, Kokonor and parts of northern Xinjiang; Xinjiang more broadly consisted mainly of military administration via garrison generals, as well as what Perdue terms 'native administration' through Turkic Muslim begs in the Turkic cities lining the Tarim Basin as well as the 'Taranchi' enclaves further north. Xinjiang would not be a full province under conventional Chinese civilian administration until 1884, half a decade after its reconquest and over a century after its initial annexation.

For further reading on the subject, see Edward J.M. Rhoads' Manchus and Han and Peter C. Perdue's China Marches West.