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The result also meant Aston Villa have qualified for Champions League for the first time since the 1982-83.
Manchester City, who are aiming for an unprecedented fourth consecutive English top-flight title, can retake the lead when they play at Arsenal's London rivals Tottenham Hotspur on Tuesday.
Pep Guardiola's relentless side, closing in on an unprecedented fourth English title in a row, has 85 points from 36 games, with Arsenal on 83 having also played 36.
Hours after Arsenal's 3-0 win over Bournemouth, Pep Guardiola's men responded with a resounding victory that pulled them to within a point of the North Londoners at the top of the standings, but with a game in hand.
Haaland made a quick impact when he entered the game as a substitute in his first action since missing two matches with a muscle injury.
Foden struck twice as Manchester City stepped up their bid for a fourth Premier League crown in a row with a ruthless 4-0 victory at Brighton & Hove Albion.
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.
Casemiro missed a spot kick for United in the shootout but they hit back and Rasmus Hojlund's penalty sent them into the final against Manchester City.
Arsenal lead on 74 points, one point ahead of Manchester City.
Silva sidefooted the ball home in the 84th minute after Chelsea goalkeeper Djordje Petrovic parried a shot from former Blues midfielder Kevin De Bruyne.