The 19-year-old's trickery, speed and skill helped the Eagles turn around a two-goal deficit to secure another Championship win on a sunny afternoon in south London.
Zaha impressed from the outset, but saw his side go behind against the run of play thanks to Chris McCann's close-range strike and an exceptional hooked effort by Martin Paterson.
Palace responded strongly and four minutes before the break Andre Moritz played through Zaha to fire home.
The same pair linked up again shortly after the restart with the same end result as Zaha fired through the legs of Burnley goalkeeper Lee Grant.
Palace looked superb in front of a bumper crowd at Selhurst Park and, despite Charlie Austin's late goal, secured all three points thanks to earlier efforts from Damien Delaney and Glenn Murray.
It was a quiet start to proceedings in south London, with shots from Kagisho Dikgacoi and Moritz the only action of note early on.
Grant soon had to react quickly to stop a Yannick Bolasie effort from looping under his crossbar before the Burnley goalkeeper got down well to stop Zaha netting from 20 yards.
A wayward Austin header was all Burnley had to show for their efforts up until the 26th minute when they broke the deadlock against the run of play.
Julian Speroni did excellently to tip a fine strike from Junior Stanislas on to the post, but McCann reacted quickly to direct home under pressure.
Less than three minutes later Burnley doubled their advantage thanks to Paterson, who exquisitely hooked the ball over Speroni from an improbable position on the right side of the penalty area.
The home support looked shocked, but Palace continued to press and, after Zaha came close and Murray had penalty appeals waved away, they pulled a goal back.
Zaha, who was impressing on both flanks, picked up a through-ball from Moritz and held off Kieran Trippier to confidently lash home in the 41st minute.
Six minutes after the restart Moritz and Zaha linked up again to level things up, with the Brazilian midfielder again playing through the home favourite to score, this time through Grant's legs.
'He's just too good for you' chanted the home fans, who would have been celebrating a third had it not been for Grant tipping wide Moritz's deflected effort.
Burnley, though, could have regained the lead had Austin not seen a shot excellently blocked by Delaney, who soon punished them at the other end.
Substitute Jonathan Williams did excellently on the right to get a ball into the box, which fell for the defender to lash in via a deflection at the second attempt.
Palace extended their advantage further in the 75th minute when Murray met Dean Moxey's low cross to direct home.
Austin pulled one back to ensure a nervy ending, but the Eagles held firm to secure a fourth successive league win for the first time since December 2007.
