They moved out of the bottom three with that point.
Ashley Barnes’ penalty put them ahead, but Villa hit back through Emi Buendia.
At Villa Park, Burnley’s Maxwel Cornet won a penalty on the stroke of half-time when he was bundled over by Emiliano Buendia.
Burnley’s only previous penalty this season, and Ashley Barnes stepped up this time, sending Emiliano Martinez the wrong way with a cool strike for his first goal since February 2021.
In the 48th minute, Buendia volleyed Villa’s equaliser from John McGinn’s cross, with Burnley protesting that a foul throw should have been given in the build-up.
Burnley finished with 10 men after Matt Lowton was sent off for a high tackle on Calum Chambers in stoppage-time.
Chelsea effectively guaranteed a third-place finish with a 1-1 draw against Leicester at Stamford Bridge.
Given their vastly superior goal difference advantage over fourth-placed Tottenham, who sit three points behind them, the Blues will not be caught.
After Chelsea’s painful penalty shoot-out defeat in the FA Cup final against Liverpool on Saturday, Arsenal’s defeat at Newcastle on Monday had secured a top-four finish for the Blues.
James Maddison gave Leicester a sixth minute lead from the edge of the area before Marcos Alonso bagged Chelsea’s equaliser, volleying home in the 35th minute.
Thursday’s drama was only an appetiser for what will be a thrilling final day of the season.
Manchester City will be crowned champions for a fourth time in five seasons if they beat Villa at the Etihad Stadium.
But second-placed Liverpool, one point behind Pep Guardiola’s team, would steal the title if City drop points and they defeat Wolves at Anfield.
In the battle to qualify for next season’s Champions League, Tottenham need a win at Norwich to be certain of pipping Arsenal, who host Everton and are two points behind their north London rivals.
Manchester United can qualify for the Europa League with a win at Palace, although they would drop into the Europa Conference League with a defeat and a West Ham victory at Brighton.