The goalkeeper’s masterclass ensured the Clarets left Villa Park with a point from a 1-1 draw after stops from John McGinn, Ollie Watkins, Danny Ings and – brilliantly – Bertrand Traore.
But it should have been better for the visitors with Tyrone Mings denying Wout Weghorst a winner before Matt Lowton was dismissed in stoppage time.
Burnley nevertheless climbed out of the bottom three on goal difference ahead of Sunday’s Premier League finale.
They host Newcastle while Leeds, a place below, travel to Brentford.
Ashley Barnes clobbered Tyrone Mings early on before the Clarets were completely caught out by a quick Villa corner but were saved when Pope turned McGinn’s curler wide.
But the visitors managed to regroup to ask questions of Villa and Matty Cash deflected Dwight McNeil’s goalbound strike over.
Burnley, though, struggled to keep the ball which left them reliant on more hopeful balls to Barnes, who was unable to link efficiently with Maxwel Cornet.
The game was becoming scrappy but Burnley still needed Pope at his best after 31 minutes.
Kevin Long failed to clear Jacob Ramsey’s pass with the ball bouncing to McGinn for the midfielder to crack a half volley from 20 yards, only to be again denied by a stretching Pope.
Burnley had shown plenty of fight, even if they were one-dimensional, and they edged ahead a minute before the break from the spot.
McNeil tossed in a free-kick, with Buendia failing with a shove on James Tarkowski, and he raced out to meet Cornet.
As the Ivory Coast international looked to skip past, Buendia recklessly swiped his legs to send him sprawling and Barnes stepped up to send Emi Martinez the wrong way for his first goal since February 2021.
It gave the Clarets a half-time lead but one they would only hold for three minutes after the break.
There was swift redemption for Buendia as McGinn spotted his run from deep and, with Burnley failing to track the Argentina international, he met the ball on the volley for it to squirm past Pope.
The self-inflicted wound hurt the visitors yet Josh Brownhill still forced a smart low save from Martinez just after the hour.
Until then Villa had bossed possession and the chance gave Burnley enough encouragement to get more of a grip of the second half – only to need Pope to ensure they claimed a point.
The goalkeeper denied Watkins and Ings drilled wide with his first touch with 13 minutes left.
But Pope produced his best with 10 minutes remaining with a superb low one-handed stop to keep out Traore’s header.
Burnley still should have won it with five minutes left after a stunning block from Mings.
Martinez initially kept out Connor Roberts with a fine reaction stop only for Roberts to tee up Weghorst and his shot was turned wide by a diving Mings.
Pope denied Ings again before Lowton was sent off in stoppage time for a foul on Calum Chambers.