Clarets boss Sean Dyche opted for Robbie Brady in place of Jeff Hendrick, while Villans manager Dean Smith made five changes to the side thumped 3-0 at Watford last time out.
Tyrone Mings, Neil Taylor, Frederic Guilbert, Marvelous Nakamba and Trezeguet were handed starts in place of Matt Targett, Ahmed Elmohamady, Conor Hourihane, Henri Lansbury and Jota.
A quiet start to the game saw both sides unable to make the most of corner-kick opportunities, while the visitors were left stunned after believing that they had taken the lead through a Grealish header.
A borderline VAR offside ruling eventually meant the strike was ruled out, with Wesley the man penalised.
The Brazilian striker responded by forcing home keeper Nick Pope into a decent save, with the Clarets gloveman also blocking the follow-up from Douglas Luiz.
Eventually, with 27 minutes on the clock, Villa did hit the front, with Wesley making up for his earlier disappointment by slotting home after chesting down a pass from the ever-involved Grealish.
And things got even better for the visitors four minutes before the half-time interval as they moved 2-0 ahead, with Grealish finding space in the home area to smack a shot past the stranded Pope.
Burnley brought on both Johann Gudmundsson and Jay Rodriguez at the interval and the latter should have pulled a goal back when he was close in, but could only turn a James Tarkowski cross-cum-shot over the crossbar.
At the other end Kortney Hause headed wide from close range from a Villa corner, while Chris Wood was unable to make the most of two decent openings for the hosts.
He did pull a goal back ten minutes from time as he headed home from an Ashley Westwood cross, with Villa keeper Tom Heaton injured and forced from the field as he tried to keep the ball out of his net.
The visitors managed to hold out despite the loss of Heaton and secured a much-needed three points in the process.