Home striker Stead netted a first-half own goal before wiping out the deficit early in the second period, and Yannick Bolasie, fresh from the bench, fired home the winner in the 82nd minute.
It gave the Robins, who had Brett Pitman sent off late on before adding a third goal through Chris Wood, breathing space with four games to play.
But for Coventry, who saw their seven-match unbeaten run come to an end, it could prove a critical blow to their survival hopes. They are the closest of the teams in the bottom three to safety, however Bristol City now look increasingly safe in 21st place.
Pitman and Stead gave Coventry goalkeeper Joe Murphy some early work to do with a couple of long-range efforts which found the target.
Ryan McGivern's 13th-minute header provided a sterner test for the goalkeeper but he did well to deny the defender after he made good contact with Albert Adomah's pinpoint free-kick delivery.
The visitors' first opportunity to apply some pressure of their own saw Gary McSheffrey whip in a corner which Dean Gerken punched clear with Cody McDonald and Clive Platt lurking.
Neil Kilkenny's superb half-volley from almost 30 yards flashed just wide as the hosts went close again.
A Robins player did get his name on the score-sheet with 33 minutes gone but Stead was credited with an own goal as he inadvertently flicked Sammy Clingan's free-kick beyond Gerken.
The home side responded well to the setback and after Cole Skuse's ambitious effort was well blocked by Richard Keogh, Kilkenny drilled another effort narrowly wide.
Despite heading into the interval behind, there were plenty of signs that Derek McInnes' men were still in the contest and within two minutes of the restart they were rewarded for their efforts.
Pitman swung in a corner, Skuse helped it on its way and Stead was there to bundle home from close range and make amends for his own goal.
With both sets of fans appreciating the huge importance of the clash there was an electric atmosphere inside Ashton Gate as the two sides went in search of a crucial three points.
The hosts were next to threaten and Richard Foster's cross was headed on target by Stead but Murphy showed quick reflexes to tip it over the bar.
After a flurry of corners Andre Amougou nodded home but referee Scott Mathieson spotted a foul by Stead on Murphy and the celebrations were cut short.
That was the lively Stead's last involvement after an eventful 66 minutes on the pitch as West Brom loanee Wood was introduced from the bench.
Bolasie came on with eight minutes to play, making an immediate and telling contribution as he turned in the box before steering a shot home from 12 yards.
Hopes of a comfortable finish appeared to have been dashed when Pitman was dismissed for a foul on Jordan Clarke but Wood's injury-time goal put the seal on a hugely important victory.