The Villans handed a league debut to left-back Matt Targett, with midfielder Conor Hourihane’s inclusion the only other change to the side beaten 3-2 by Arsenal.
The visitors named the same starting XI as the one that began the 2-0 home victory over Norwich City a week ago.
Early on, Targett delivered a fine cross to set up John McGinn, but his header drifted over the crossbar when he might have done better, while at the other end Chris Wood was unable to make a connection when he was well placed.
Hourihane then saw a free-kick stay out after hitting a post before Dwight McNeil rattled the crossbar for the visitors with a set-piece delivery of his own.
McGinn had the ball in the net 25 minutes in although a lengthy VAR intervention eventually ruled the effort out for offside.
However, Villa were 1-0 in front 12 minutes before half-time as Anwar El Ghazi finished superbly to smack a Frederic Guilbert ball in from the right past the stranded Nick Pope.
Early in the second half, Wesley needed treatment following a clash with Ben Mee, with the Brazilian eventually able to continue. However, Targett was not so lucky and had to be replaced by Neil Taylor just past the hour mark at Villa Park.
Soon afterwards Burnley were level after Erik Pieters fired over a cross from the left and Jay Rodriguez made no mistake with a powerful header past Tom Heaton.
Villa responded and McGinn restored their lead 11 minutes from time as he met a far-post cross from Trezeguet with a left-foot volley.
The lead lasted less than two minutes, though, as former Villan Matt Lowton crossed and Wood was on hand to score with a firm header to ensure the spoils were shared after an end-to-end encounter.