In their first away game of the season, the women's Super League leaders overcame a stubborn Reading side to win.
Stina Blackstenius put the visitors in front with a clinical finish inside the penalty area.
The Swede almost doubled the lead early in the second half when she hit the woodwork with a glancing head after the same player had shot.
Kim Little's penalty was saved by Reading's goalkeeper, who was making her top-flight debut, after she was shown a yellow card for handling the ball.
The second-worst side in the WSL, Woodham, was on the verge of tying the game when they were denied by a brilliant save from the keeper, who set a new WSL record with her eighth successive clean sheet.
After three games, Manchester United are at the top of the table with a goal difference of four points.
Reading, who set a new attendance record for their women's team, have lost their first three games.
Few would have predicted that Reading would pose a threat after one of the first attacks they faced.
Despite a lack of service for willing runner Dowie in attack, the organisation and energy of Kelly Chambers' players became a feature of a determined display in which they so nearly peggedArsenal back, and showed a marked improvement on their pair of four goal defeats in the fixture last season.
Most Royals fans wouldn't have expected a positive result but this was a performance to suggest that they can get points from their next two games.
Chambers said that if they don't take the chances, it could hurt them.
We didn't take those opportunities. We have the positives and we need to build on the performance.
Before the start of the game, the boss said that the two players could not be expected to play like the injured centre-backs.
Since December 2021, when the England captain has been involved, the English club have lost more matches than they have in the 20 they have played.
When Cooper broke free inside the penalty area and drew a vital stop from Zinsberger, who was the hero for more than one occasion, it was Wubben-Moy who was to blame.
There were enough opportunities to score a second goal. It's always going to be a bit more tense when you only have one goal.
It was well deserved and we did a good job of controlling it. Everyone understood that we had to go up in intensity once we got out of their pressure.
Eidevall is certain to learn more about his backline's ability to stand up to sterner tests when he travels to Lyon for the first leg of the group stage.
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