But the win was clouded in controversy after the home side had strong appeals for a penalty turned down in injury time when Danny Brown's free-kick seemed to hit the arm of Mark Convery.
Crawley surrounded the referee but their arguments fell on deaf ears as the official blew for full-time.
Hollins, fresh from receiving the joint Conference Manager of the Month award, was left reflecting on his first defeat in seven games after some slack marking at a corner gifted City the lead on the hour mark.
Andy Bishop rose highest to head on to the post and, with keeper Phil Smith grounded, Dudgeon was left with the simple task of tapping into an empty net from two yards.
By that time, Crawley had squandered a golden chance and had a goal ruled out.
Neil Jenkins should have scored after only four minutes when he was picked out by Steve Burton but his low side-footed effort was saved brilliantly by Chris Porter.
Crawley's top scorer Daryl Clare then thought he had scored soon after when he nodded home a free-kick by Simon Wormull but his effort was chalked off for offside.
York could have added another nine minutes from time when Bishop was released on goal by a poor back pass from Tony Scully, but he uncharacteristically wasted the gilt-edged opportunity as Smith raced off his line to make the save.
Crawley surrounded the referee but their arguments fell on deaf ears as the official blew for full-time.
Hollins, fresh from receiving the joint Conference Manager of the Month award, was left reflecting on his first defeat in seven games after some slack marking at a corner gifted City the lead on the hour mark.
Andy Bishop rose highest to head on to the post and, with keeper Phil Smith grounded, Dudgeon was left with the simple task of tapping into an empty net from two yards.
By that time, Crawley had squandered a golden chance and had a goal ruled out.
Neil Jenkins should have scored after only four minutes when he was picked out by Steve Burton but his low side-footed effort was saved brilliantly by Chris Porter.
Crawley's top scorer Daryl Clare then thought he had scored soon after when he nodded home a free-kick by Simon Wormull but his effort was chalked off for offside.
York could have added another nine minutes from time when Bishop was released on goal by a poor back pass from Tony Scully, but he uncharacteristically wasted the gilt-edged opportunity as Smith raced off his line to make the save.