The barbarians who sacked Rome were themselves Christians by this time. During the Second Sack of Rome (the first happened well before the founding of the Empire, when Rome was still just another city of italy) the sacking Goths even decreed that churches were safe from the sacking, as were anyone taking refuge within one. The Catholic Church was never really threatened by the barbarians.
And, of course, remember that only half of Rome fell. The Eastern Roman Empire was stronger than ever at this time, in control of the Holy Land and four of the five Bishoprics that made up the Pentarchy (Rome, Constantinople, Jerusalem, Antioch, and Alexandria). Christianity flourished, and is the basis for modern Eastern Orthodox Christianity, a major force in a European history.