Despite the visitors dominating the early possession, it was City who carved out a couple of chances. Clayton Donaldson beat his marker 25 yards out but shot wide, and then turned provider with a cross which Andy Bishop headed straight at keeper Steven Drench when well placed.
City keeper Chris Porter and defender Evan Horwood collided when going for a cross and Porter needed lengthy treatment before continuing, but Horwood suffered double vision and was replaced by Dave Merris.
Morecambe then created two excellent chances from similar positions on the left edge of the box with both Michael Twiss and Danny Carlton beating Porter only to see their low drives slide agonisingly inches wide.
Donaldson showed great skill flicking the ball over Chris Blackburn and firing in a dipping volley from an acute angle which was cleared for a corner.
City scored the opening goal on 42 minutes with a cracker from Mark Convery. Striker Donaldson once again showed great skill beating his marker in a confined space before sending in a teasing cross which was half-cleared and fell kindly for Convery to smash in on the half-volley from 18 yards.
Morecambe responded well after the break and Curtis was denied an equaliser by a superb diving save from Porter. From the resulting corner City failed to clear their lines and John Hardiker's thundering cross-shot struck Curtis and ended in the back of the net.
Gary Thompson almost put the Shrimps ahead with a delicate chip that beat Porter but struck the bar. Then Emmanuel Panther for York and Thompson again for the visitors went close from distance as both sides tried in vain for a winner.