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Ten-man Chelsea beat Brighton & Hove Albion 3-2 at Stamford Bridge while fourth-placed Aston Villa needed a late Ollie Watkins goal to scrape a 2-2 draw at Bournemouth.
Trent Alexander-Arnold drilled in an 80th-minute equaliser with his right foot to deny City a 24th successive home win in all competitions.
Cole Palmer scored a late penalty to make it 4-4 at Stamford Bridge in London.
The result left Spurs second in the standings behind Manchester City.
City thoroughly dominated shell-shocked United in the second half.
Arsenal came into the match having not beaten City in the league since 2015, a run of 15 games -- their longest winless streak against a single opponent in league history.
Manchester United's campaign of woe continued with a 1-0 home loss to Crystal Palace but there were no such troubles for Arsenal who, along with Spurs, closed the gap to leaders City to one point with a rampant 4-0 win at Bournemouth.
In all, referee Anthony Taylor doled out 11 yellow cards along with Rodri's red, with seven of the yellows coming in a bad-tempered second half as Forest tried to leverage their extra man, but to no avail.
City started their title defence by coming from behind to beat Red Star Belgrade 3-1 at home.
Manchester United tasted a 1-3 defeat at Old Trafford while City won by the same margin against the Hammers.