Goals from Anthony Martial and a Bruno Fernandes penalty either side of a strike by Aston Villa's Bertrand Traore were enough to bring Man United up to 33 points from 16 matches. Liverpool, who play Southampton on Monday, are still in first due to a superior goal difference.
The teams traded half-chances in an entertaining end-to-end opening 45 minutes before Martial headed home from an Aaron Wan-Bissaka cross on a rapid-fire counter-attack to stake the hosts to a 1-0 lead at the break.
Traore levelled the score shortly after the restart, firing home from close range after Jack Grealish picked him out with a low cross into the area.
United peppered Martinez's goal as the second half progressed, with Fernandes and Marcus Rashford forcing saves and Pogba firing just wide of the Villa goal.
The hosts saw out the result and will look to continue a good run of form when they face Manchester City in the Carabao Cup semifinals on Jan. 6 (watch live on ESPN+ at 2:45 ET).
Wan-Bissaka, who constructed the home side's opening goal for Martial, was upbeat after United saw their unbeaten run stretch to 10 games.
"The win means a lot, it shows where we've come from, from the start of the season," the 23-year old told the BBC.
"We're just taking it game by game. It was a late start for us, we were in catch-up mode, but we're here now. No excuses."
Information from Reuters was used in this report.