Arsenal looked on course to leapfrog United as Beth Mead finished from close range with four minutes of a tight London derby remaining.
Excellent work from Vivianne Miedema led to Mead smashing the ball home at Meadow Park – but the drama was not done there.
The visitors struck back just four minutes later as Pernille Harder’s cross was deflected by Lotte Wubben-Moy, and the ball looped into the back of the net.
The Blues had a glorious chance to settle the game late on as Sam Kerr raced through but her chipped effort rolled narrowly wide of the post.
The result leaves Man United top of the table with 17 points from seven games, with Arsenal one point behind.
Chelsea are three points off United, but have a game in hand and would go top if they won it by virtue of their superior goal difference.
Additional reporting by PA.