Arsenal went top of the standings with the win because of 0-0 draws in matches involving Liverpool and Chelsea.
Liverpool goalkeeper Pepe Reina saved a penalty kick in a 0-0 draw at Portsmouth. Chelsea had a goal disallowed in its 0-0 draw with Blackburn.
Emmanuel Adebayor scored twice at White Hart Lane, with Cesc Fabregas putting Arsenal ahead for the first time in the 80th minute. Tottenham took the lead in the 15th through Gareth Bale, with Adebayor levelling in the 65th and adding Arsenal’s third goal in the 90th.
"We had that midfield strength and that togetherness pushed us through," Arsenal manager Arsene Wenger said. "It needed something special as Tottenham played well."
Manchester United also needed a late winner, with Nemanja Vidic heading into goal in the 83rd for the Red Devils to beat Everton 1-0.
Also Saturday it was: Birmingham 1, Bolton 0; Sunderland 2, Reading 1; West Ham 3, Middlesbrough 0; and Wigan 1, Fulham 1. On Sunday, Manchester City hosts Aston Villa.
Arsenal leads with 13 points, followed by Liverpool, Manchester United and Chelsea with 11 after a two-week international break. West Ham and Everton have 10 points.
"We feel we grow from game to game and become stronger and stronger," Wenger said.
Tottenham has not beaten Arsenal in any competition since November 1999, when it won a league match 2-1 at White Hart Lane. Spurs only have one win this season from six games.
"We should have killed them off but we didn’t take our chances," Tottenham manager Martin Jol said. "We took the game to them and created chances. But they were clinical and that was the difference."
Spurs took the lead when Bale curled a 25-yard free kick around the wall and past Arsenal goalkeeper Manuel Almunia.
Arsenal went on the attack, with Abou Diaby hitting the crossbar and Tottenham goalkeeper Paul Robinson making two saves from Adebayor in the first half.
Fabregas levelled the scores with a long-range bending strike into the top corner before Adebayor made it 3-1 with a volley.
.
Andriy Shevchenko started for Chelsea for his first appearance this season, replacing the injured Didier Drogba, but he missed two chances in the first half.
Blackburn goalkeeper Brad Friedel made an acrobatic save in the 27th to deny Michael Essien from scoring with a rising half-volley. Chelsea had the ball in the net early in the second half through Salomon Kalou but it was ruled offside.
Blackburn midfielder Robbie Savage forced Chelsea goalkeeper Petr Cech into a one-handed save in the 76th. Friedel also stopped a header from Shevchenko in the 84th minute, but the Ukraine striker’s boot hit the head of defender Christopher Samba, who was taken off the field on a stretcher.
At Goodison Park, Vidic headed into goal after taking a corner from Luis Nani.
The Red Devils had the better chances throughout the game, with Patrice Evra and Cristiano Ronaldo — returning from a three-match suspension — missing shots. Carlos Tevez had an attempt blocked by Everton defender Joleon Lescott, and Michael Carrick and Vidic also came close in the first half.
Early in the second half, Everton striker Andrew Johnson flicked a corner from Mikel Arteta that was cleared off the line by Paul Scholes. United defender Rio Ferdinand made a late block to deny Everton substitute Victor Anichebe an equalizer.
"Nemanja had the determination and courage to score in their penalty box at a great time for us and Rio showed real determination in his area to get a block in when we needed to see it through," United manager Sir Alex Ferguson said.
Everton fielded loan signing Stefan Wessels in goal after Tim Howard dislocated his finger while playing for the United States.
Liverpool could not break down Portsmouth but kept its unbeaten away record. Reina dived to his right to stop a penalty kick from Nwankwo Kanu in the 32nd minute. Kanu had earned the penalty after Liverpool defender Alvaro Arbeloa tugged his shirt.
Peter Crouch had a seventh-minute chance saved by Portsmouth goalkeeper David James and sent a back-heeled shot just over the crossbar early in the second half. Andriy Voronin hit the crossbar for Liverpool in the 54th and just missed a getting onto a loose ball right near the goal in the 86th.
Portsmouth had good chances through John Utaka and Sulley Muntari.
"If you take a point from the likes of Liverpool, Manchester United, Chelsea or Arsenal you are never going to complain because they are top teams," Portsmouth manager Harry Redknapp said.
West Ham beat Middlesbrough for its first home win of the season with goals from Lee Bowyer, Dean Ashton and an own-goal from Luke Young.
Wigan drew 1-1 with Fulham after a penalty kick from Jason Koumas in the 80th minute. That cancelled out an 11th-minute goal from Fulham midfielder Clint Dempsey.
Sunderland earned its second win of the season with a 2-0 win over Reading. Kenwyne Jones hit a long-range shot past Reading goalkeeper Marcus Hahnemann in the 29th minute.
Ross Wallace doubled the lead in the 47th minute, but Dave Kitson pulled a goal back for Reading in the 85th.
Birmingham defeated Bolton 1-0 through a 37th-minute goal from Olivier Kapo.