The Clarets were leading at half-time through Ashley Barnes' 44th minute penalty.
But Villa levelled two minutes into the second-half, courtesy of Emiliano Buendia's well-taken volley.
Matt Lowton was sent off late on, but Burnley held on to secure a point.
James Tarkowski returned to captain Mike Jackson's starting line-up for the fixture in the Midlands, replacing Lowton in defence.
First-half
Burnley had some defending to do in the opening minutes. Charlie Taylor blocked Buendia’s shot in the box, while Nathan Collins was forced to clear the follow-up cross for a corner.
Up the other end, Barnes’ flick-on was met by Dwight McNeil to weave into the box, but Luca Digne charged down his effort.
Pope was forced into action, pushing John McGinn’s curling strike from distance over the crossbar after Buendia’s short corner.
The Clarets had a chance of their own on 20 minutes. McNeil’s goal-bound drive following a corner was headed over his own crossbar by Matty Cash.
On the half-hour, Pope made an excellent save, keeping out McGinn’s fierce half-volley with a strong left-hand.
Ollie Watkins later sent a header off target from Cash’s ball, as Villa pushed for an opener.
Burnley were awarded a penalty with 44 minutes played when Paul Tierney pointed to the spot after Cornet was fouled by Buendia.
Up-stepped Barnes who converted his right-footed spot-kick into the bottom left-corner, sending goalkeeper Emiliano Martinez the wrong way.
Second-half
Steven Gerrard's side equalised inside two minutes of the restart. McGinn's pinpoint lofted pass into the box dropped to the unmarked Buendia to volley right-footed which beat Pope.
Buoyed by their equaliser, Villa threatened the Clarets goal, as Buendia went close again.
With 64 minutes gone, Cornet's cross was headed out by Tyrone Mings into Josh Brownhill's path for the midfielder to hit a low drive that Martinez saved.
VAR intervined midway through the half to deny the Clarets a penalty, which adjudged Digne not to handle in the box under pressure from Barnes.
Wout Weghorst was introduced as a substitute, replacing Ashley Barnes for the closing 20 minutes.
Pope had to be at his best to tip Watkins' powerful effort over for a corner, prior to Bertrand Traore side-footing Digne's pullback wide.
Former Burnley striker Danny Ings, on as a substitute, almost edged Villa ahead with a clean strike from McGinn's lay-off which went narrowly past Pope's bottom left-corner.
Pope was superbly positioned to keep Burnley in the game, thwarting Traore after he met Calum Chambers' cross.
Jackson made two changes in quick succession late on, with Aaron Lennon and Lowton coming on for McNeil and Cornet.
In a frantic end to the clash, both sides traded chances. Taylor's cross found Connor Roberts who was denied by Martinez's save on the line. The Welshman reacted quickest to the loose ball, teeing up Weghorst, with the forward denied a tap-in from Mings' last-ditch challenge.
Shot-stopper shines again in sparkling performance
Lennon's shot was saved by Martinez, while Pope denied Ings, Coutinho and McGinn. Lowton was sent-off for a challenge on Chambers, as the Clarets navigated the four minutes of additional time to earn a point.
Next-up for Burnley is a final day game against Newcastle United at Turf Moor, kick-off 4pm.
Aston Villa: Martinez, Cash, Chambers, Mings ©, Digne, Ramsey (Coutinho '70), Douglas Luiz, McGinn, Buendia, Watkins (Ings '77), Chukwuemeka (Traore '56)
Subs not used: Olsen, Sanson, Young, Nakamba, Iroegbunam, Feeney
Burnley: Pope, Roberts, Long, Collins, Tarkowski ©, Taylor, Brownhill, Cork, McNeil (Lennon '80), Cornet (Lowton '84), Barnes (Weghorst '70)
Subs not used: Hennessey, Bardsley, Stephens, Thomas, Dodgson, Costelloe
Goals: Barnes, '44, Buendia, '47
Referee: Paul Tierney
Attendance: 40,468
Sent-off: Lowton, '90.