Rooney headed United level in the 78th minute at Old Trafford and Javier Hernandez nodded in the winner two minutes later.
Although the comeback will ease the immediate pressure on manager David Moyes, United provisionally remains eighth, eight points behind leader Arsenal.
Arsenal won 2-0 at Crystal Palace, with Mikel Arteta scoring a penalty before being sent off, while Olivier Giroud headed in the second goal.
Liverpool produced the most convincing victory of the day, with Luis Suarez scoring a hat trick in a 4-1 victory over West Bromwich Albion that lifted the team to second place, two points behind Arsenal.
Everton is provisionally in fourth place with a 2-0 victory over Aston Villa in the latest installment of the most played fixture in English football history, coming on the day the Football Association celebrated its 150th anniversary.
Southampton moved provisionally third with a 2-0 win over Fulham after Rickie Lambert and Jay Rodriguez scored for the hosts. The Saints are arguably the biggest surprise team this season in the Premier League — not counting Man United's unexpected struggles.
The last two decades in English football have been dominated by Manchester United, but in their first season since 1985-86 without Alex Ferguson in charge, the team is far from convincing.
Moyes' side has already dropped 13 points this season after it looked like the ninth league game would witness more misery.
Peter Crouch bundled the ball into the net after just four minutes after Stoke left back Erik Pieters beat United defender Chris Smalling to cross to the striker.
Stoke, managed by former United star Mark Hughes, remained a threat throughout the first half.
Robin van Persie levelled for United in the 43rd after goalkeeper Asmir Begovic failed to fully save Rooney's header.
But Stoke was back in front before the break after Marko Arnautovic sent a free kick into the top corner of the net.
Boos rang out around Old Trafford at half time, and United took until the 78th to level again with Rooney flicking a header into the net from Van Persie's corner.
Two minutes later came another headed goal, this time from Hernandez and the game was won for United.
"Stoke have not scored many but they have not conceded many either, so we knew we would have to work hard after going behind," Moyes told the BBC. "I was worried it wouldn't come — especially when we were 2-1 down. But I think we deserved it in the end."
But it is Arsenal setting the pace in its hunt for a first league title since 2004.
At Crystal Palace, Arteta opened the scoring from the penalty spot in the London derby two minutes into the second half after Serge Gnabry was fouled by Adlene Guedioura.
But Arsenal had to play the last 25 minutes with 10 men when Arteta was adjudged by referee Chris Foy to have brought down former Arsenal striker Marouane Chamakh when he was still about 35 yards away from the goal.
Goalkeeper Wojciech Szczesny protected Arsenal's lead with a diving one-handed save to tip away a long-range shot by Mile Jedinak. But there would be no equalizer for Palace, which is next from bottom and searching for a new manager after Ian Holloway left on Wednesday.
Olivier Giroud headed in Arsenal's second in the 88th after connecting with Aaron Ramsey's cross.
"We were mature, organized and didn't become nervous and we got the second goal," Arsenal manager Arsene Wenger said.
Arsenal is showing it is coping fine without Suarez, who was pursued unsuccessfully during the summer transfer window.
At Anfield, though, there was no stopping Suarez in front of goal on Saturday.
The first goal came after 12 minutes, with Suarez picking up possession on the right and skipping past Yacob before nutmegging Jonas Olsson and netting with an angled shot.
The Uruguay striker was on target again with a header five minutes later from distance, and he completed his hat trick by nodding in another 10 minutes into the second half.
James Morrison grabbed one back for West Brom from the penalty spot before a spectacular long-range lob from Daniel Sturridge completed Liverpool's convincing win.
At Villa Park, Christian Benteke squandered an early chance to put the hosts in front when Everton goalkeeper Tim Howard saved his penalty after eight minutes.
Benteke's Belgium teammate Romelu Lukaku then put Everton ahead in the 69th minute with his fifth goal in five games, and Leon Osman added the second in the 81st to hand Villa its fourth home loss of the season.
Everton only has one loss in nine games and provisionally climbed above Chelsea, Manchester City and Tottenham, who all play Sunday.
Norwich remains in the relegation zone after drawing 0-0 with Cardiff, which is just a point higher up.Suggest a correction