The Eagles made hard work of it in the first half, with Dele Adebola fetching some cheer from the empty terraces - just 4,901 showed up - with his first goal of the season coming on 20 minutes.
After the break Palace came out with all guns blazing and the Robins were sunk after conceding three goals within the first ten minutes of the restart.
Skipper for the night Hayden Mullins headed home Wayne Routledge's cross on 47 minutes, Tony Popovic's far-post header arrived seven minutes later from Danny Butterfield's corner and Dougie Freedman finished off a slick move for Palace's fourth on 55 minutes.
The Eagles took a breather for a while but continued on the goal spell after 73 minutes with a well placed header from Adebola. Richard Walker added to Cheltenham 's woes by diverting sub Julian Gray's cross into his own net before Freedman finished the rout with a minute to go.
Cheltenham left the field dazed after holding their own in the first half. which saw Tony Naylor missing the best chance after only ten minutes.
Palace Boss Trevor Francis said after the match: "It was a game of two halves. In the first half we were bad but in the second, we looked a different team.
"It's what a manager looks for in a team, we showed good width and young Wayne Routledge excelled again with some good crosses.
"Adebola deserved his two goals as he has been working really hard without success. After his goals tonight you could see a huge weight lifted off his shoulders."
Cheltenham boss Graham Allner said: "The three goals straight after half time killed us.
"It's not usually like a Cheltenham team to cave in like that. We held our own until the goal and I thought our best football came ten to fifteen minutes before half time.
"At half time I thought we could get something out of the game. I wasn't happy with some of the goals we conceded and we have now conceded 11 in two games.
"Hopefully come Saturday we can get it back to square one and we have three days to sort it out."