Having won Palace's player of the month for February, the inform midfielder forced the opener for the Eagles after Burnley defender Phil Bardsley turned in a shot from Schlupp to score an own goal.
The Eagles put in another superb away performance to win at relegation rivals Burnley and take another step towards securing their top-flight status.
Both sides went into the game level on 30 points and Palace made the early breakthrough as Jeffrey Schlupp's cut-back cannoned off Phil Bardsley's thigh into his own net.
Michy Batshuayi doubled Palace's lead just after the break, sweeping in a low ball by fit-again full-back Aaron Wan-Bissaka.
Palace continued to keep it tight at the back while Wilfried Zaha grabbed the goal his work-rate and class deserved, tormenting the Burnley defence before firing low past Tom Heaton.
Ashley Barnes pulled one back late on while Burnley substitute Peter Crouch almost set up a nervy finish, but Palace held on to claim their third straight away game in all competitions.
The victory also lifted the Eagles up to 13th in the Premier League, putting eight points between themselves and the relegation zone, while Burnley dropped one place to 16th, five points clear of third-bottom Cardiff.