Anton Stach’s stunning long-range free-kick gave the visitors the lead just past the half-hour mark, only for late substitute Tammy Abraham to bundle home an equaliser for Villa two minutes from time.
It meant Leeds were denied a first victory away from home since a 3-1 win over Wolves at Molineux on September 20, but they did stretch their unbeaten league run to three matches, with this result leaving them six points clear of the drop.
The visitors' confidence was clear to see through the opening stages and they forced the first big chance in the 14th minute.
Dominic Calvert-Lewin latched onto Jayden Bogle's delivery but was smothered by the onrushing Villa keeper Emiliano Martinez.
Leeds made all the running and made their first half dominance count through a moment of brilliance.
Ollie Watkins brought down Ethan Ampadu over 30 yards from goal and Stach faked to put a ball in the box.
But instead, he whipped a beautiful free-kick past the diving Martinez and into the back of the net.
Villa were in danger of shipping more goals in a lacklustre performance and United could have had a second when Calvert-Lewin teed up Ilia Gruev, who arrowed his shot marginally over the crossbar.
Ilia Gruev (right) spent much of the game engaged in a fierce midfield battle, but he almost scored a cracking goal too. (Image: Bruce Rollinson/PA Wire)
Leeds were clinging onto their lead heading into the break as Villa sustained their best spell of pressure.
Watkins put a shot straight at Karl Darlow when he ought to have done better, before Amadou Onana's point-blank effort was saved by the Leeds keeper minutes later.
Villa began to turn the screw following the introduction of Jadon Sancho on the hour mark and he started to orchestrate the Villa attack.
The best chance fell at the feet of Tyrone Mings at the edge of the box, but his effort was blocked.
The home side thought they were on level terms midway through the second half after Emi Buendia brilliantly curled a shot on to the post, with Watkins on hand to tap it home but the assistant referee's offside flag quickly cut their celebrations short.
Lukas Nmecha almost doubled Leeds' advantage with his first contribution off the bench when he dived to meet the cross of Jayden, Bogle but Martinez got across well to stop his header.
And that Martinez save proved to be crucial when a late corner was poked home by the thigh of Abraham, whose effort went above the leaping Calvert-Lewin on the line to earn Villa a point.