The hosts dominated without being at their best but were thwarted by the Clarets, who climbed out of the Premier League relegation zone.
Anwar El Ghazi and Kortney Hause hit the woodwork while Matt Lowton cleared a deflected strike from Ahmed Elmohamady off the line.
Burnley's threat was limited - Chris Wood having their best chance saved - but Sean Dyche's side are now unbeaten in three after a tricky run against Everton, Arsenal and Villa.
The Clarets rode their luck as Villa missed the chance to make serious ground on the Premier League's pacesetters.
They remain 11th, four points adrift of fifth after failing to score - despite having 27 shots.
The hosts started well and Bertrand Traore flashed a drive wide before Nick Pope turned Ollie Watkins' cross away.
Burnley's inability to retain possession aided the hosts but the Clarets suddenly sparked in to life after 19 minutes.
Dwight McNeil's driving run opened up space and he found Jay Rodriguez to cut back to Ashley Westwood for the former Villa midfielder to curl just wide.
Villa responded and only Charlie Taylor's excellent last-ditch challenge thwarted Traore at the far post before Emi Martinez's flying save clawed Wood's header out at the other end.
The game had opened up, although Villa remained on top, and Burnley survived a scare after half an hour.
Elmohamady's drive took two deflections, the final one off Watkins, and Lowton - who spent three years at Villa Park - headed off the line.
Pope then frustrated them further nine minutes before the break when he turned El Ghazi's 25-yard free-kick onto the bar.
Villa continued to probe and, after Pope gathered Watkins' weak shot, Hause's header hit the bar from Traore's corner three minutes before half-time.
The hosts had got progressively closer but their 16 shots were the most by any Premier League side in the first half of a game without scoring in over a year.
El Ghazi curled wide after Robbie Brady's loose pass 11 minutes after the break as the hosts looked for the breakthrough which never came and the winger also saw a shot deflect over.
Villa pressed as Burnley looked to cling to a point and build on their fine win at Arsenal at the weekend and El Ghazi and Tyrone Mings were both off target while Martinez grabbed Ben Mee's snapshot.
Jack Grealish also drove over with a minute left but Villa were forced to settle for a draw.
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