It seemed United were destined to go five games without a win as familiar defensive frailties allowed Villa to go 2-0 ahead in the space of six first-half minutes following John McGinn's unchallenged free-kick and a goal from an unmarked Leander Dendoncker.
United, however, staged an impressive rally in the second half, with Alejandro Garnacho scoring twice before a long-awaited first Premier League goal from Rasmus Hojlund's secured a win that lifted the club innto sixth place two days after Jim Ratcliffe's upcoming investment in the Red Devils was announced,
Victory over Villa was only United's second win in seven games during December, but manager Ten Hag has been without the likes of Lisandro Martinez, Casemiro, Mason Mount and Tyrell Malacia, all due back in January along with Victor Lindelof.
"So often we have to change the team," said Dutch boss Ten Hag. "You don't get the routines. We know football is about solid performance and consistency and we know we have to make a step there, but I'm sure when we have more players available in the key positions we will get more consistency."
Marcus Rashford made his first United start since December 2nd, with the England forward having initially lost his place to Garnacho before illness delayed his return.
"I think he played very well," Ten Hag said of Rashford. "He was ill and also he had one or two games where we preferred to play Garnacho on the left side but he deserved it. There is internal competition.
"(On Tuesday) we played Alejandro on the right, that can be a solution, it can be fluid, but Rashford can also play on the right side."
Villa were all set to end the game level on points with leaders Liverpool before United's second-half fightback, with manager Unai Emery urging the Birmingham club to put the result behind them.
"Move on, be demanding," he said. "I told the players that the first part of the season was fantastic but I want more. If they want more we have to work hard to get it.
"My mentality now is to focus on Saturday (at home to Burnley) and prepare the match as well as possible. We want to be better tomorrow than today."
