Zaha helped to create both goals and he got his own name on the scoresheet with a superb solo strike in the 76th minute to ensure Palace continue to look up the Premier League table, although Ashley Barnes hit a late consolation for Burnley.
Palace took the lead in the 15th minute, Bardsley turning the ball into his own goal after Jeffrey Schlupp fired a Zaha cross back into the danger zone.
Chris Wood scored an early winner in this fixture last season and almost replied quickly, hitting the chest of goalkeeper Wayne Hennessey after linking with strike partner Barnes.
Batshuayi doubled Palace's advantage in the 48th minute. Zaha was involved again and a superb driven cross from Aaron Wan-Bissaka was smashed home by the striker.
Scott Dann's brilliant block denied Barnes a reply for Burnley before Schlupp went close with a long-range effort, with Batshuayi's poor touch meaning he failed to capitalise on a Ben Mee error.
Any slim hopes of a comeback were seemingly extinguished in fine style as Mee and Charlie Taylor were turned inside out by Zaha before the Ivory Coast international rifled home a wonderful finish.
Barnes netted for the fifth time in six Premier League games to complete the scoring, nodding in Ashley Westwood's cross, with Hennessey making a fingertip save to deny substitute Peter Crouch in added time as Burnley belatedly battled back.
What does it mean? Burnley slip back into trouble
A run of eight games without defeat - their longest such sequence in the Premier League - seemed to have hauled Burnley away from relegation danger. But successive defeats to Newcastle United and Palace have left them nervously looking over their shoulders again.
Roy Hodgson has suggested Palace should be targeting a top-half finish and, with safety almost assured, a strong finish should see them continue to move up the table.
Zaha's hot streak continues
Palace's star man Zaha is in great form, having contributed four goals and an assist in his previous five Premier League appearances. While he will not be credited with assists for either of Palace's first two goals, they both came from his prompting and his own clinching third was superbly taken.
Taylor out of his depth
All three Palace goals came from their right with Burnley defender Taylor having a game to forget. The left-back needed more support from those in front of him but he was beaten too easily and too regularly, although he did cause the Eagles some problems when he got forward.
What's next?
In-form Palace have a derby clash with Brighton and Hove Albion at Selhurst Park on Saturday. The following day, Burnley go to Premier League title hopefuls Liverpool.
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