Simply put, the persecutions by the Roman Empire were not comprehensive enough to check the rise of Christianity as a movement and religion.

There were certainly quite a few cases of persecution against Christians in the Roman Empire, for all sorts of reasons: mob-led action from local communities (this one was particularly prevalent), local governors getting paranoid about revolts, philosophical and religious differences between Roman and Christians (some of the most obvious ones being that God is the highest good and not the Empire, and monotheism vs. polytheism). For all these reasons and more, early Christians had to be wary of acts of misguided violence against them.

But ultimately these acts of violence were outdone by the religion's spread via missionary work. Christ was said to have told his followers to spread his teachings all throughout the world, and so many people spread out to establish churches in the Greek area, the Asia Minor area, and even within the heartland of the Empire itself. Being so close to the Mediterranean, with its importance as a hub for a large part of the world, was quite helpful for nascent Christianity. There were centers of Christian learning in cities such as Alexandria and there are references in documents to bishops and leaders of the faith even if the faith wasn't near the juggernaut we know it as today.

Regardless, though, Christianity was illegal in the eyes of the Roman State until Emperor Constantine's reign. Therefore this made them "others" of sorts with all the negative side-effects that come from being an "other". However, and this is key to your question: the persecutions weren't really ORGANIZED until about the 200s. Nero's infamous torching of Christians wasn't something that spread all around the empire, but was confined to Rome. Many emperors were hesitant to make specific laws against Christianity despite its illegal status. So it largely depended on how cruel the local leaders and populace wanted to be to Christians. This uncentralized persecution allowed the spread of Christianity to far outpace instances of persecution against them.

So by the 200s, there were likely millions of Christians in the Empire despite these persecutions. Christian churches were popping up more and despite prejudice from the locals and the illegality of their faith, there really were many Christians able to survive in the Empire. These churches could indicate that some Christians were even able to acquire wealth despite their persecuted status. Some orthodoxy and scripture was established as well as hierarchy within the Church. Their increased visibility would cause them some more problems, however.

One of the biggest problems was likely Diocletian, an emperor in the late 200s that oversaw an actual systemic, legislative persecution of Christians. He was a staunch Roman conservative and did not like what he saw as a cult spreading throughout the Empire. Among other things, he denied Christians many legal rights, purged his army of them, slaughtered Manichaeans, destroyed churches, imprisoned bishops and priests...the list goes on. This was probably the darkest time for the early centuries of Christianity as a faith.

Not long after this dark time for Roman Christians, though, did probably the most important events for their social status occur. The Peace of Galerius sought to usher in tolerance for all faiths, but it was somewhat milquetoast and persecution still continued in some ways after that; the Edict of Milan in 313, however, was the hole-in-one; Emperor Constantine ended all legal persecution of Christians, said they could practice their faith without consequence, restored property and provision to previously persecuted Christians, and much more. Constantine himself also favored the faith, and whether he was a full convert before death or follower is up to debate, but he did several things to increase legitimacy of Christianity, some of which being funding churches, declaring Sunday a day of rest, and using Christian symbols on coinage.

Christians' evangelism and endurance in the face of persecution (especially in the late 200s) were some of the reasons they survived, but Constantine's edict and his favorable actions to Christendom are possibly the most important things to how Christianity was able to legally thrive and indeed dominate - Diocletian may have done some damage to the faith but Christianity was here to stay after that.

You can probably see from here how the Church gets more and more powerful. In 380, Nicene Christianity becomes the official religion of the Empire. From here on, more and more power and wealth is consolidated by bishops and church leaders even after West Rome collapses; and ultimately they go on to become the most powerful faith in the world.

A quick, minor point to clarify for those confused, OP made the common and understandable mix-up of "Disciples" and "Apostles". Not all of the Disciples died martyrs, but all but one of the Twelve (John. Judas too, but he kind of doesn't count, I guess) are said to have. The question is of course quite fine aside from that minor point of clarification.