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.
Arsenal made it five consecutive wins in the Women’s Premier League and six in all competitions with a convincing 3-0 victory over Brighton.
Stina Blackstenius opened the scoring in the 12th minute with a shot into the top left-hand corner of the net before Caitlin Foord doubled their advantage with 10 minutes to go in front of a record crowd of 4,921 at a sold-out Broadfield Stadium.
Frida Maanum sealed the three points in stoppage time with her first goal of the season after being teed up by Cloe Lacasse, leaving the second-placed Gunners three points behind leaders Chelsea in the table.
Bristol City twice came from behind to earn a 2-2 draw with fellow strugglers Everton at Walton Hall Park.
Martina Piemonte put the hosts in front just five minutes into the contest but Amy Rodgers slammed home from inside the area just before the interval to bring the scores level.
Everton retook the lead in the 57th minute courtesy of Megan Finnigan and looked on course for only their second win of the WSL season when Amalie Thestrup dinked home from close range with eight minutes remaining to earn rock-bottom City a share of the spoils.
Tottenham extended their unbeaten run to six matches with a 1-1 draw against Leicester at the King Power Stadium.
Leicester went ahead in the 18th minute when Janice Cayman received a pass from Hannah Cain before firing home but were pegged back when Celin Bizet latched onto a through ball and finished coolly.
Viviane Asseyi’s penalty put West Ham a goal to the good but Villa turned the game on its head through Anna Patten and Adriana Leon.
West Ham thought they had snatched a point when Lisa Evans drilled home from outside the area with 10 minutes to go.
But, the decisive moment came two minutes into added time when England star Daly smashed into the top corner to snatch all three points for Villa.