On a bitter cold afternoon, City soon gave the visiting fans some festive cheer when they took the lead just four minutes in.
Luke Wilkshire curled in a free-kick from the left and centre-back Tony Butler glanced home a header past the despairing dive of County goalkeeper Neil Cutler.
Warren Feeney responded with a blocked shot for Stockport, then on 11 minutes John Hardiker's powerful header was frantically cleared off the line by Wilkshire.
The basement battlers deservedly got back on terms on 19 minutes when Feeney flicked the ball over Butler and fired home with some aplomb from 20 yards.
Clearly keen to impress new boss Chris Turner, Stockport kept up the tempo and Feeney went close again with a left-footed drive.
The impressive Jon Daly was having a great game up front for Stockport, but he blazed over from a great position on 40 minutes.
Bristol did have the last word of the first half, however, when County old boy Tony Dinning saw his thumping drive superbly tipped over by Cutler.
The visitors went back into the lead ten minutes after the restart. Leroy Lita's weak angled shot was only palmed away by Cutler, straight into the path of Heffernan who couldn't miss from five yards.
Stockport's best effort in what was a disappointing second half for them came when Feeney saw his looping header superbly pushed away for a corner by City goalkeeper Steve Phillips.
Left-back Danny Adams saw a similar header scrambled away for another corner late on.