The former Tractor Boys striker won the match with 60 seconds remaining to give the side rock-bottom of the npower Championship a huge three points in their battle against relegation.
The visitors led at half-time thanks to the recalled Daryl Murphy but, just two minutes after the break, substitute Steven Davies levelled with a fine finish.
Former Nottingham Forest man David McGoldrick played a part in the game's opening goal on the half-hour mark. He flicked the ball to the recalled Murphy, who drew Tom Heaton in before coolly firing home his sixth of the season.
City, who have conceded more goals on home soil than any other team in the division, then came very close to letting in the 35th of the campaign eight minutes before the break when Lee Martin somehow missed the target from six yards after being found by McGoldrick's cross.
Just two minutes after Sean O'Driscoll's first half-time team talk at Ashton Gate, the hosts were level. Louis Carey's raking ball found Paul Anderson and the former Liverpool winger's reverse pass found Davies, who finished from a tight angle to draw the home side level.
Heaton made a big double save with six minutes to go to deny Aaron Cresswell and then Lee Martin as he looked certain to score.
And just when it looked like the sides would have to settle for a point, debutant Brendan Moloney stood up a cross which Davies knocked down into the path of Stead and the striker kept his cool to earn a vital win.