United trail their traditional rivals on goal difference with both teams on 33 points from 16 matches ahead of their clash at Anfield on Jan. 17. Liverpool travel to Southampton on Monday while United visit Burnley on Jan. 12.
United stretched their unbeaten league run to 10 games as Bruno Fernandes capped another fine individual performance with a penalty after Bertrand Traore had cancelled out Anthony Martial's opener for the home side.
United right back Aaron Wan-Bissaka, who constructed the home side's opening goal for Anthony Martial, was upbeat with the winners of a record 20 league titles looking to mount a credible challenge this term.
"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."
The hosts had the upper hand in the first half and Martial forced an acrobatic save from Emiliano Martinez before he buried a superb diving header past the Villa goalkeeper after Wan Bissaka's fine cross in the 40th minute.
But Villa's joy was short-lived as Fernandes drilled his penalty into the bottom right corner three minutes later after Paul Pogba went down under a challenge from Douglas Luiz, with the spot-kick given after a VAR check.
United pressed on and came close through Pogba as well as Fernandes and Marcus Rashford before being forced to defend desperately in the dying minutes, with centre back Eric Bailly putting in a crucial stoppage time block.