The pick of the Minstermen's three goals came from Paul Robinson, making his debut following his recent move from Tranmere Rovers.
The visitors dominated for the first 20 minutes and created a number of chances, and David Stockdale had to be alert to save well from volleys fired in by Allan Tait and moments later Maurice Harkin.
Neil Jenkins then went on a solo run and shot just wide before Stockdale saved again from Daniel Marney's low shot.
City bossed the later stages of the half with Robinson and Darren Dunning both trying their luck from 25 yards.
Kevin Donovan's trickery on the right flank started to cause the visitors problems and from one of his crosses Paul Groves saw his header cleared off the line by Joe Dolan.
Another superb move down the right involving Donovon and Groves ended with Dave Merris heading back across goal for Andy Bishop to nod inches wide.
City went ahead after 40 minutes when Lee Grant headed Donovan's free-kick home from seven yards.
Within 60 seconds Dolan had an equaliser disallowed for pushing before Tait saw his effort ruled out for offside.
Crawley began well after the restart with Simon Wormull a major influence in midfield his astute pass set up Marney for the equaliser on 51 minutes.
Moments later Wormull's vision set Jenkins free but keeper Stockdale saved bravely at his feet.
City went ahead after 69 minutes with a superb goal from debutant Robinson.
Having dispossessed Ryan Palmer by the corner flag he cut between two defenders and curled an exquisite 20-yard effort into the back of the net via the far post.
Bishop wrapped up the three points by heading home Merris' cross five minutes from time.