The Red Devils were awarded a penalty in the 20th minute as Alex Greenwood handled inside the area, with Katie Zelem slotting home to put them a goal up.
City were not behind for long and turned the game on its head with two goals inside a minute as Jill Roord slotted home the equaliser before Lauren Hemp curled in superb fashion from outside the area to make it 2-1.
The visitors ensured the three points as Khadija Shaw closed down a back-pass to Mary Earps, with the goalkeeper's clearance bouncing off Shaw and going into the net to rubber-stamp the triumph.
Despite Laia Aleixandri's second bookable offence for a foul on Lucia Garcia, City held on to move above United in the WSL.
Elsewhere, Arsenal secured their sixth win in all competitions with a convincing 3-0 victory over Brighton thanks to goals from Stina Blackstenius, Caitlin Foord and Frida Maanum.
Bristol City twice came from behind to earn a 2-2 draw with fellow strugglers Everton at Walton Hall Park while Tottenham's unbeaten run continued with a 1-1 draw against Leicester.
Rachel Daly scored in stoppage time to help secure Aston Villa their second win of the season with a 3-2 victory over West Ham.
Additional reporting from PA