Substitute Paul Anderson looked to have secured a 90th-minute point for Bristol City when Albert Adomah's free-kick cannoned back off the bar, but midfielder McCann headed home Kieran Trippier's raking set-piece in the sixth minute of time added on.
Charlie Austin escaped the attentions of the City defence to give Burnley a ninth-minute lead, scoring for the eighth consecutive match to equal a club record set by Ray Pointer in the 1958/59 season.
The 23-year-old's 13 goals in this period compared to Pointer's 12 put him one ahead in the record books - a position Burnley were not in for very long on the pitch thanks to Steven Davies' free-kick.
City were on top for the rest of the opening period but were left to rue their failure to make the pressure count as Burnley raced clear with a quickfire double.
Handball in the area by Mark Wilson allowed Austin to make it 14 in eight and Martin Paterson's deflected strike found the net moments later.
Sam Baldock's penalty set up a grandstand finish and both sides obliged, leaving Burnley caretaker manager Terry Pashley with two wins from as many games in charge.
McCann's lofted pass from deep caught out the Robins backline and Austin stole in to round goalkeeper Tom Heaton and seemingly inevitably open the scoring.
Burnley's enterprising start was emphatically checked in the 17th minute when Michael Duff fouled Davies just outside the D.
Complaints from the visitors that Davies had gone to ground a little easily were compounded when the same player picked himself up and arched the free-kick over the defensive wall and in
Davies tried to repeat the trick six minutes later but fired over and at the other end Paterson - into the starting line-up because Ross Wallace suffered an injury in the warm-up - might have extend Heaton further from Austin's deep cross after the half-hour.
It was a brief moment of respite for Burnley, who found themselves largely on the back foot after City's equaliser, but it remained all square at half-time after Liam Fontaine fizzed a volley over from Stephen McManus' header.
A quiet start to the second half exploded around the hour as Pashley's men took first control of the contest.
Paterson's header from Junior Stanislas' cross struck Wilson on the hand, referee Lee Collins pointed to the spot and Austin duly obliged to register his 18th goal of the season after 59 minutes.
It was three a minute later when Paterson's back-post effort from McCann's cross took a wicked deflection off Richard Foster to leave Heaton with no chance.
City were handed a lifeline in the 75th minute when Burnley centre-back Michael Duff was adjudged to have fouled McManus from Wilson's corner and Baldock slammed the penalty into the centre of the goal.
Adomah fired against the underside of the bar after being fouled by Brian Stock to leave Anderson with a simple finish - a luxury McCann was not afforded when he angled home at the death to condemn City to a third straight loss.