The Clarets took the lead from the penalty spot at the end of the first half when Emiliano Buendia tripped Maxwel Cornet inside the box.
Ashley Barnes, without a goal in 15 months when netting against Fulham at Turf Moor, confidently dispatched the spot kick having sent Emiliano Martinez the wrong way.
The hosts equalised within the first few minutes of the second half when Matty Cash’s foul throw was missed by the officials, and then ignored by VAR, prior to Buendia atoning for his earlier error and steering John McGinn’s cross past Nick Pope.
Mike Jackson’s men could have won it late on when Roberts, who had been foiled by Martinez, squared the ball to Wout Weghorst, but the substitute’s close range effort, with the goal gaping, was blocked spectacularly by Tyrone Mings.
The result saw the Clarets climb out of the bottom three heading into the final game of the season.
Here are the player ratings.
1. Nick Pope 9
Outstanding. Might have done better with the goal, but it's hard to criticise the England international after another wonderful display. He made two stunning saves to keep out McGinn in the first half and then trumped those with a sensational stop to keep out Traore's header in the second half. Fantastic! Photo: Stu Forster
2. Connor Roberts 6.5
Villa targeted his side of the pitch in the first half, with Digne delivering the ball from deep, but the Welshman refused 18-year-old Chukwuemeka access on the outside. Could have reacted better to McGinn's cross when trailing Buendia, but set up substitute Weghorst late on for what looked like a certain goal. Photo: Alex Livesey
3. Charlie Taylor 7
Made a number of blocks inside the box, including one from Buendia in the opening seconds of the game. Battled exceptionally well against McGinn and Cash, chasing the latter back to the byline countless times to block crosses. His own delivery much improved from Sunday at Spurs, including the assist for a late Roberts chance. Photo: Richard Heathcote
4. Kevin Long 6.5
The centre back never lets Burnley down when called upon. Spent the first half clearing the ball from inside his own penalty as the Clarets forced Villa into wide areas. Up against it a bit more when the hosts played through the middle, but stuck to his task to keep things as tight as possible. Photo: Alex Livesey