City's unbeaten league start looked under threat when Steven Caulker headed Tottenham in front, but goals from Sergio Aguero and Edin Dzeko clinched a 2-1 win.
As the champions fell two points behind United, Chelsea dropped a point further back in third after being held to a 1-1 draw at Stamford Bridge.
Chelsea captain John Terry marked his return from a racism ban by heading in his 50th club goal, but the defender was later carried off with a right knee injury and Luis Suarez equalized in the second half.
In Sunday's other match, Kevin Nolan scored against former club Newcastle to give West Ham a 1-0 victory at St. James' Park.
In Manchester, City came from behind for the fourth time in the league this season — a day after United recovered to beat Aston Villa 3-2.
The arrival of Dezko from the bench made the difference for City, as the striker scored his seventh goal of the season in the 88th minute after being picked out in the penalty area by Edin Dzeko.
"We felt we needed an injection," said David Platt, a coach under manager Roberto Mancini. "We needed a little bit more height to get a platform to hold the ball up.
"We also know he is a goal threat and he is fresh. He has scored lots of goals this season and since he has been here he has scored some heavy goals."
Slack goalkeeping by Joe Hart allowed Tottenham to take the lead in the first half as Caulker's header from Tom Huddlestone's free kick was helped into the net.
But Caulker was caught out in the 65th as Aguero went past him before equalizing with a shot in the bottom corner.
Dezko's winner left Tottenham in seventh place, three points out of the top four Champions League places but a point ahead of Arsenal going into Saturday's north London derby.
Chelsea is the leading team in London, but has now drawn two matches and lost to United since last winning in the league.
Terry had given Chelsea the perfect start as he returned from his four-match ban for racially abusing Queens Park Rangers defender Anton Ferdinand.
The centre back headed in Juan Mata's corner in the 20th minute at Stamford Bridge after being left unmarked by the Liverpool defence.
But 15 minutes later, Terry fell awkwardly on his right knee after a collision with Liverpool striker Luis Suarez and was carried off. The full extent of the injury won't be known until Terry undergoes medical scans on Monday.
Without Terry's presence at the back, Liverpool found the space to equalize in the 73rd minute.
Jamie Carragher was able to cleverly head captain Steven Gerrard's corner onto the far post where the unmarked Suarez nodded in his 11th goal of the season.
"Suarez is an amazing player. He is really clever and the key player for us," Liverpool defender Jose Enrique said.Suggest a correction