Shrewsbury capitalised on their extra man advantage after Jarvis saw red halfway through the second half.
Neither side were able to break the deadlock in the first half, despite a fairly dominant performance from the visitors.
The scoreline stayed that way even with Shrewsbury knocking on York's door, with several shots on goal saved by York goalkeeper Michael Ingham.
Jarvis picked up a straight red on the 64th minute to make Russ Wilcox's first game as manager that little bit more difficult.
Collins condemned Wilcox to defeat on his first game as York manager as his right-footed shot from outside the box flew into the bottom corner.