Goalkeepers Jordan Pickford and Emiliano Martinez contributed significantly to Aston Villa head coach 's first goalless draw in 97 Premier League matches.
Clement Lenglet is set to stay at Aston Villa despite a host of top European clubs chasing the defender.
Lyle Foster was on the score sheet for Burnley but Aston Villa moved level on points with Premier League leaders Liverpool as Douglas Luiz's late penalty secured a 3-2 home win.
Rasmus Hojlund's long-awaited first Premier League goal completed a stunning comeback as Manchester United recovered from 2-0 down to beat high-flying Aston Villa 3-2 at Old Trafford.
Brentford and Aston Villa have been charged by the Football Association (FA) over the scenes late in their Premier League clash on Sunday.
Mikel Arteta was seemingly aiming to sidestep another Football Association charge as the Arsenal boss refused to be drawn on two contentious calls in their Premier League defeat at Aston Villa.