Anton Stach’s rocket free-kick looked to have earned the Whites a rare three points on the road until Villa levelled on 88 minutes.
Leeds have conceded 14 goals in the last 15 minutes of league games this season and still have only won one away game.
Whether that statistic comes back to haunt Leeds at the end of the season remains to be seen but a point at a club chasing a Champions League spot was welcome.
Daniel Farke has made his side hard to beat since the turn of the year as third-placed Villa discovered.
He said: “We are disappointed to only take a point… but my warriors deserve all the praise. We were really good on the ball. With confidence we could have taken more advantage.
The Yorkshire side deservedly went ahead when the impressive Stach scored with a brilliant free-kick which swerved and dipped with great power from 30 yards into Emi Martinez’s top corner on 31 minutes.
It was a bolt from the blue with just about everyone, Martinez included, expecting the German simply to whip it in to the box.
Martinez had earlier denied Dominic Calvert-Lewin with a brave block after the Leeds number nine – a menace throughout – had been played in by Jayden Bogle’s pass over Villa’s backline.
United were getting plenty of joy on the break while Unai Emery’s men came up with a well-drilled tough defence.
It was only at the end of the first-half that Leeds goalkeeper Karl Darlow was brought into serious action, first tested by Leon Bailey before spreading himself to block an Amadou Onana effort from close range.
It took an hour for Villa, energised by substitutes Ross Barclay and Jadon Sancho, to really get going.
An Emiliano Buendia shot hit the post but Ollie Watkins tapped in the rebound from an offside position but Villa were building up a head of steam.
Although Leeds were coming under increased pressure they still posed a threat when they managed to get forward.
Minutes after coming on Lukas Nmecha’s diving header was blocked by Martinez who later kept out a Calvert-Lewin shot.
Villa fans, seeing a home game with a Saturday 3pm kick-off for the first time in a year, were baying for the introduction of Tammy Abraham.
They got their wish on 75 minutes and the striker, who rejoined the Villains from Turksih club Besiktas in an £18.2m deal last month, delivered the goods.
Ezri Konsa won a backpost header from a corner for Abraham to loop the ball high into the off his thigh to break Leeds hearts.
With West Ham held at home to a goal-less draw by Bournemouth, Leeds, on 31 points, remain six points ahead of the relegation zone with 11 games left, the first of those being against title-chasing Manchester City at Elland Road on Saturday tea-time.