Uncharacteristic defensive frailties by Chelsea allowed Tottenham to beat its London rival for the first time in almost five years, with Harry Kane scoring twice.
Never before, across Jose Mourinho's two spells as manager, had Chelsea conceded so many times.
"We made some defensive mistakes," Mourinho said. "We had some individual problems with the defensive structure."
As did Man City, but the champions recovered after squandering a two-goal lead for the second time in a week, preventing a repeat of the 2-2 draw against Burnley on Sunday.
It showed just how important it was for City late on New Year's Eve to secure the 36-year-old Lampard until the end of the Premier League season rather than see him return to sister club New York City FC to start the Major League Soccer campaign in March.
With Chelsea and Man City level on points, goal difference and goals scored, only alphabetical order keeps Chelsea on top.
Arsenal and Liverpool are struggling to make the top four.
Liverpool threw away a 2-0 lead to draw 2-2 with bottom-place Leicester and stay eighth, while Arsenal missed a chance to dislodge Southampton from fourth place by losing 2-0 at the surprise high-flier.
Southampton is only a point behind Manchester United after the third-place team drew 1-1 at Stoke, with Radamel Falcao recovering the point.
All three teams in the relegation zone drew. Crystal Palace, which is on the verge of completing the managerial signing of Alan Pardew from Newcastle, drew 0-0 at Aston Villa. Burnley drew 3-3 at Newcastle, which is yet to appoint Pardew's successor.
West Bromwich Albion confirmed Tony Pulis as the new manager earlier Thursday, and he was at Upton Park to see his team draw 1-1 at sixth-place West Ham to stay a point above the relegation zone.
Everton is surprisingly in the bottom half of the standings — three points above the danger zone — after Hull's 2-0 victory inflicted a fourth straight loss. Queens Park Rangers' survival hopes were helped by a 1-1 draw against Swansea.
No game could match the wild drama produced at White Hart Lane.
Chelsea led briefly. Eden Hazard was denied by the post, but Oscar sent the ball back across the face of goal and Diego Costa was primed to tap into the net in the 18th minute.
But Kane levelled on the half-hour, facing little resistance before powering the ball past goalkeeper Thibaut Courtois, and Tottenham led from the 44th. After Nacer Chadli hit the post, Danny Rose resisted Gary Cahill's sliding challenge to bury the follow-up in the net.
The lead was extended before the break from the penalty spot by Andros Townsend after Kane was tripped by Cahill. And Kane was on target seven minutes into the second half after cutting through the Chelsea defence again.
Tottenham's defence was less impressive.
On the hour, Federico Fazio was dispossessed near the halfway line, Hazard seized the ball, and played a one-two with Cesc Fabregas before dispatching the ball low past goalkeeper Hugo Lloris.
Although Chadli's deflected shot beat Courtois in the 78th, Chelsea captain John Terry pulled another back for Chelsea.
Chelsea and City have the same goal difference, and both have 44 goals in 20 games.
Lampard's goal for City — a header from Gael Clichy's cross — was his 176th in the Premier League, taking fourth place on the division's all-time leaderboard.
After coming off the bench, his 76th-minute winner came after City goals by Yaya Toure and Stevan Jovetic were cancelled by former players Jack Rodwell and Adam Johnson.
Arsenal could find no way back against Southampton after Sadio Mane scored from a tight angle Dusan Tadic capitalized on a defensive error.
"It was a self-inflicted defeat," Arsenal manager Arsene Wenger said. "We made the mistakes at the back."Suggest a correction