The striker netted after 54 minutes on Tuesday night before adding a second four minutes later to send the npower Championship side into the second round for the fourth year running, where they face a home tie against Wigan.
Crawley, who were guilty of missing an array of chances in the first half, more then held their own against a side two divisions higher and could have opened the scoring five minutes before Zaha's opener but Tyrone Barnett fired against the post.
In the first meeting between the teams, Crawley, who recalled fit-again striker Matt Tubbs in their only change from the weekend win over Torquay, seized the initiative early on and almost took the lead inside seven minutes but Lewis Price acrobatically turned Josh Simpson's volley around the post.
Charlie Wassmer had a shot deflected wide before Palace, who made nine changes from Saturday's win over Hull, came back into the game and should have opened the scoring but on-loan QPR defender Peter Ramage could not find the net from six yards.
Town looked slightly vulnerable at the back and Darren Ambrose should have taken full advantage of a defensive mix-up but he fired straight at Scott Shearer when clean through.
Steve Evans' team started off where they left off in the first half and, on 48 minutes, Barnett crashed his shot against the upright.
However, against the run of play, the hosts went in front after in clinical fashion as Zaha fired past Shearer from 18 yards.
From the next attack, Zaha converted his and the Eagles' second from an Andrew Davies knock down.
The Red Devils continued to attack, though, and Sergio Torres had a shot blocked by Davies before substitute Scott Neilson fired wide.
In the closing stages, Zaha fired over with his hat-trick in sight but Palace held on to claim their fourth consecutive victory.