Javier Hernandez's double and Ron Vlaar's own goal completed the turnaround that took United four points clear of Chelsea at the top of the table. Chelsea hosts Liverpool on Sunday.
"We did put pressure on ourselves today but we came through it," United manager Alex Ferguson said. "The lesson is we need to do better defensively. I admire the tenacity of the performance, though. When they need to do it, they do it."
At Arsenal, Mikel Arteta had a penalty saved in stoppage time as the draw prolonged the north London team's worst start to a league season since Arsene Wenger took charge in 1996.
Olivier Giroud and Lukas Podolski put Arsenal ahead, but Fulham drew level before halftime as Dimitar Berbatov and Alex Kacaniklic scored from free headers. Berbatov put Fulham ahead from the penalty spot after the break but Giroud headed Arsenal level with his second goal.
Everton beat Sunderland 2-1 to stay fourth and ahead of West Bromwich Albion, which won at Wigan by the same scoreline.
Queens Park Rangers is bottom after losing 1-0 at Stoke, but Southampton moved a point further up after drawing 1-1 with Swansea, while Reading drew 0-0 with Norwich.
United had looked set to lose its third match of the season when Andreas Weimann scored twice.
In first-half stoppage time, he was true with a rising first-time shot from just inside the penalty area, and in the opening five minutes of the second half, Weimann smashed the ball in after being set up by Gabriel Agbonlahor.
But Hernandez, who replaced Ashley Young at halftime, began United's comeback in the 58th, slipping the ball past Brad Guzan after latching onto Paul Scholes' pass.
Five minutes later, United was level after Hernandez's shot from Rafael da Silva's cross deflected in off Vlaar past Guzan.
After United twice hit the woodwork, Hernandez sealed the three points for United in the 88th by heading in from Robin van Persie's free kick.
Hernandez, though, was claiming a hat trick.
"I think I hit the target and the defender touched the ball," the Mexico striker said of United's second goal. "That's the decision of the referee but I've just seen the replay and that was definitely hitting the target."
There was as much drama earlier at Emirates Stadium where Arsenal threw away a two-goal lead just like against Schalke in Tuesday's 2-2 draw in the Champions League.
Theo Walcott set up the 11th-minute opener for Giroud to head in, and Arsenal went further ahead when Arteta cut the ball back past Chris Baird to Podolski six yards from goal and the Germany forward swept the ball home in the 23rd.
Fulham's fightback began in the 29th when Berbatov rose unmarked to meet Bryan Ruiz's corner and he sent the downward header with ease over the line. From another free header, Kacaniklic equalized after meeting Podolski's cross.
After the break, just as Arsenal seemed to be taking control, Arteta tangled with Ruiz and referee Phil Dowd awarded a penalty kick. Berbatov coolly slotted into the bottom left from the spot in the 67th.
The lead lasted only two minutes, with Giroud heading home after an initial effort hit the post and fell to Walcott, who returned the ball to the France international.
"It was frustrating because we had the opportunities to win," Wenger said. "We were caught in a bit too comfortable of a position because we did the most difficult thing — we scored two goals to go 2-0 up. At that moment we lacked urgency defensively."
At Goodison Park, David Moyes' 400th league match in charge of Everton was marked with a victory that ended a run of four straight draws.
Adam Johnson grabbed his first goal for Sunderland on the stroke of halftime, but Marouane Fellaini and Nikica Jelavic scored within three minutes after the break to clinch Everton's win.
West Brom is behind Everton only on goal difference.
James Morrison headed West Brom in front at Wigan after Chris Brunt's deflected cross looped over Wigan goalkeeper Ali Al Habsi.
Wigan captain Gary Caldwell sent the ball into his own net after being hit by Billy Jones' cross. Arouna Kone pulled one back for Wigan just before halftime.
QPR's miserable start to the season continued as Charlie Adam scored his first goal since joining Stoke in August, netting at the start of the second half after Jon Walter nodded a free kick onto the Scotland midfielder.
QPR is still searching for its first win.
Southampton looked on course for just its second win since gaining promotion when Morgan Schneiderlin headed the south-coast side ahead in the 64th.
But the Saints conceded their 29th goal in 11 matches when winger Nathan Dyer netted from the edge of the area for Swansea.