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City's Bernardo Silva and Mateo Kovacic both misfired in the shootout to put Madrid in the driving seat.
Arsenal's defeat, their first in 12 league games, sees them remain second, two points behind City and level on points with third-placed Liverpool with six games left to play.
Pep Guardiola's men have 73 points with six games remaining in a rip-roaring three-way title race, two points ahead of both Arsenal and Liverpool, who can climb over City with victories in their respective games on Sunday.
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.
With seven games remaining in the neck-and-neck title race, Liverpool and Arsenal have 71 points with the Gunners leading on goal difference. Holders Manchester City are third on 70 points, while United are sixth.
Arsenal's 10th win in their last 11 league games put them on 71 points from 31 games.
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.
Arsenal survived a penalty shootout to scrape past Porto and reach the quarter-finals of the Champions League for the first time in 14 years.
Manchester United fanned the embers of their top-four hopes with a 2-0 win against visiting Everton.