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It was the sixth time Arsenal have scored five or more in a Premier League game this season without conceding and in such a tight race they now have a huge goal difference advantage which could prove crucial to the outcome.
City's Bernardo Silva and Mateo Kovacic both misfired in the shootout to put Madrid in the driving seat.
Federico Valverde salvaged a draw for Real late on after City had taken a 3-2 lead with sumptuous strikes from Phil Foden and Josko Gvardiol following a Ruben Dias own goal and Rodrygo strike that gave the hosts a half-time lead in Spain's capital.
Mikel Arteta's side moved into first place with 68 points from 30 games although Liverpool, who have 67, can move back to the summit with a home game against Sheffield United.
Liverpool claimed a 2-1 comeback victory over Bright.n & Hove Albion.
Mikel Arteta's team are third with 61 points after 27 games, a point behind Manchester City.
But the 30-time Portuguese champions will travel with hope.
It was a sublime display by Arsenal who wiped off a goal difference deficit to champions Manchester City who they joined on 52 points, although they stay in third place courtesy of scoring fewer goals.
Fourth-placed Aston Villa suffered a 3-1 home defeat by Newcastle United, Crystal Palace came from behind to beat visiting Sheffield United 3-2 and struggling pair Fulham and Everton fought out a goalless draw at Craven Cottage.
While Liverpool's 7,000 travelling fans celebrated, Arsenal's left the Emirates Stadium chuntering about the need for a new striker in the transfer window.