Dixon led with his studs in a vengeful tackle after being caught by O'Brien's raised boot moments earlier, the Motherwell winger's gesture of putting the ball out of play failing to avert the retribution.
United boss Peter Houston felt the dismissal was harsh, but both managers agreed the incident helped swing the match in the home side's favour.
Well drew with 10-man sides in Aberdeen and Hamilton, as well as nine-man United before Brown took charge and their failure to break down opponents reduced in numbers saw their unbeaten SPL run halted at 12 matches.
Brown said: "I wish he hadn't been sent off because we had been doing well and United got great energy from that and they got a terrific lift because they felt aggrieved."
While Dixon's dismissal was a key moment, the pivotal period came in a six-minute spell before half-time when, as Brown admitted, Motherwell "gifted" United two goals.
The visitors had chances to take the lead after the sending-off but the unmarked Jamie Murphy headed wide from eight yards and John Sutton was flagged offside after deflecting Keith Lasley's shot home.
United took the lead in the 38th minute when David Goodwillie edged home Andy Webster's header and Prince Buaben soon found an empty net after John Ruddy failed to control a backpass.
Motherwell struggled to penetrate the United defence after that and Danny Swanson broke clear to net a third in the 72nd minute.
The Fir Park men slipped four points behind United, and will now take an even keener interest in next month's Scottish Cup semi-finals, with only a Dundee United-Celtic final allowing fifth place to net a Europa League place.
Brown said: "Give credit to Dundee United, they played exceptionally well, particularly when they were reduced to 10 players.
"It's taken us 12 games to lose three goals, so you have to give credit to United.
"We've got to prove next week that we can respond to that. We play Falkirk at home and we've got to get back to winning ways.
"That's probably the biggest travelling support we've had and I feel sorry for them because they came away up here full of expectation.
"We've got to do our best in the remaining games to apologise effectively for that performance. We will certainly be back in the remaining games of the season and won't be as softly beaten as that."
Houston praised his team's workrate as they followed up their midweek cup victory over Rangers with their fourth consecutive home win.
The Tayside club can now further consolidate third spot with victory against Hibernian at Easter Road on Wednesday.
Houston, who made five changes, said: "One thing abut the United players this season, they work their socks off and they are all in it together.
"I've said a few times that we've got a good squad and we had to prove we had a good squad with (Jon) Daly and (Morgaro) Gomis and the like left out.
"But the players I brought in gave me a headache for Wednesday night and long may that continue."