Eddie Howe's side were two goals to the good just three minutes later thanks to a superb chip from Martin Paterson.
Palace responded well and had a lifeline before half time through Zaha, who finished neatly after a pass from Andre Moritz.
The two combined to great effect again on 52 minutes to level the scores, with Zaha finishing from inside the area to score his second.
The turnaround was complete just after the hour when Damien Delaney latched onto a cutback from substitute Jonathan Williams.
A shellshocked Burnley conceded a fourth 15 minutes from time when Glen Murray produced a simple finish for his seventh goal in as many matches.
Charlie Austin kept the game alive with ten minutes to go with his 15th goal of the season, making him the country's top goalscorer.
Palace held on to take Dougie Freedman's men takes up into the Championship playoff places.