The sides went into the game with 19 draws between them and the brilliance of Nick Colgan in Stockport's goal promised to add another to each side's total.
Twice in a lacklustre opening period his dives kept his side in the game and he was not to blame when Peacock finally beat him in the 69th minute.
With his back to goal the striker was allowed to turn and get in a right-footed shot into Colgan's right corner.
Stockport manager Sammy McIlroy put the blame squarely on his defence: "I can't mock them for all round effort but one sloppy moment cost us points and we are back in the bottom four."Peacock, who had gone six games without a goal, said: "My turn caught the keeper wrong-footed but it was hardly something I expected to do either. I was expecting support from our midfield but it just wasn't there."That comment broadly summed up a game which was sucked into the mud. City lacked any inspiration from their midfield while Stockport, who always got men behind the ball, could offer only patchy support to their front runners.
Many of their moves rested on the speed Andy Welsh summoned up down the muddy left flanks.
Behind him the pace of full-back Danny Jackman was often tested to the full by Bristol's right flank man Christian Roberts.
The Welsman's speed promised to unlock the game but luckily for Stockport his delivery seldom came up to scratch.
There was just one booking when Roberts caught Jackman high up on the left leg. It was an accidental challenge for which he quickly apologised.