We propose a new microscopic mechanism to explain the unusually fast fusion process of carbon nanotubes. We identify the detailed pathway for two adjacent (5,5) nanotubes to gradually merge into a (10,10) tube, and characterize the transition states. The propagation of the fused region is energetically favorable and proceeds in a morphology reminiscent of a Y junction via a zipper mechanism, involving only Stone-Wales bond rearrangements with low activation barriers. The zipper mechanism of fusion is supported by a time series of high-resolution transmission electron microscopy observations.