Danny Allsopp should have doubled the lead in the 20th minute. City broke away and Stuart Green played a great through ball for Allsopp, but he took the shot early and hit it straight at Macclesfield 'keeper Steve Wilson.
Minutes later the home side very nearly punished City, but Burgess was on hand to head unmarked Graham Potter's effort off the line.
They came desperately close with another corner four minutes before half time. The ball bounced off the crossbar from George Abbey's effort, then Junior Lewis was fortunate to block Martin Carruthers on the rebound. The half time whistle was a welcome sound for City as it ended a succession of Macclesfield corners.
The Silkmen carried on where they left off, and after a Paul Harsley shot flashed wide they were level in the 64th minute. Michael Welch flicked on a cross to Matthew Tipton to score.
Welch headed a good chance over, and although City got back into it after bringing Jon Walters and Ryan France off the bench, they couldn't find a way through to goal.
City line-up (4-4-2): Myhill; Thelwell, Joseph, Delaney, Dawson; Green, Ashbee, Lewis, Elliott (France 72); Allsopp (Walters 72), Burgess.
Un-used subs: Musselwhite, Hinds, Forrester.