Ryan Clarke produced a brilliant penalty save to help Oxford United reel off their fourth straight win and stay top of the Blue Square Premier.
Leading by an own goal early in the second half, United conceded a spot kick on 72 minutes when Damian Batt was adjudged to have brought down Paul Lorraine, and was yellow-carded for it.
Danny Kedwell struck the kick well enough but Clarke beat it out to his right.
The breakthrough came four minutes into the second half, and although it was an own goal, it was a great move from the U's which set it up.
Adam Murray and James Constable combined on the left, Dannie Bulman played it out to his right for Chapman, who saw Damian Batt sprinting on the overlap.
Batt crossed low to the near post and as Murray and Jay Conroy tried to connect, the ball appeared to come off the Dons defender and deflect into the goal.
The first half had been a tight and scrappy affair but United showed excellent defensive qualities all over the pitch, closing down to deny the home side space and forcing them into mistakes.
Chris Wilder began with a more compact formation, recalling Luke Foster in place of the injured Rhys Day, and keeping to his promise to mix up his squad by giving a first start in midfield this season to Chapman, and another opportunity to Marcus Kelly on the left wing.
New signings Ross Perry and Steven Kinniburgh were on the bench, along with Matt Green.
The Kingsmeadow Stadium was packed to the rafters and it was a good playing surface for the players.
Dons striker Luke Moore had the first real opening of the game on 12 minutes, although United had controlled the play until then.
It came when Batt missed his kick as a cross came over from the left, but Moore blazed well over.
Chapman hit a free-kick straight into the wall, and a little later, Kevin Sandwith hit another free-kick well over the bar.
Constable fired over a dangerous low cross from the right which just eluded all of his teammates, and Batt saw a deflected shot saved.
The best opportunity of the first half fell to Chapman, who sent a glancing header from Bulman's cross arc just past the far post.
Bulman had been terrific in the centre of the park, but Murray wasn't able to get much joy with Steven Gregory shadowing him wherever he went.
Foster showed good pace at the back when he overtook Kedwell as he raced towards goal.
Just after Oxford had taken the lead, Wimbledon upped the tempo of their game as they tried to get back on level terms, and they enjoyed a good spell of possession without finding a way through.
Conroy, hoping to make amends for his own goal, struck a long-range drive which Clarke saved on his knees.
And Clarke was fortunate that Kedwell miscontrolled a dropping ball and the ball bounced straight into the keeper's arms.
Midway through the second half, Wilder brought on Green to give Constable some support, with Kelly making way, and Perry came on for his debut with five minutes to go to try to shore things up.
Dons gave it a real go and created chances near the end.
Moore sent a header into Clarke's arms, and sub Jon Main was inches away from turning in an equaliser, although a linesman had his flag up for offside in any case.
In injury time, both Constable and Green missed glorious opportunities to extend United's lead on the break, but in the end it didn't matter.
Oxford Utd: Clarke; Batt, Foster, Creighton, Sandwith; Chapman (Perry 85), Bulman, Clist, Kelly,(Green 67), Murray (Potter 90), Constable. Subs not used: Turley, Kinniburgh.
Ref: Steve Creighton (Reading)
Att: 4,304 (820 from Oxford)