After joining from Leyton Orient on a three-year deal on Sunday, Odubajo struck the winner for his new club as the Championship outfit saw off their Premier League opponents.
Following a relatively uneventful first period, the game came to life in the second half with four goals in 10 minutes shortly after the restart.
Jonathan Douglas and Andre Gray put the home side two goals to the good, but Glenn Murray and Marouane Chamakh hit back for Palace.
Odubajo then stepped up to win the game 15 minutes from time as Tony Pulis' men suffered their first defeat of pre-season.
The hosts made the brighter start, Stuart Dallas and Alan Judge both bringing saves out of Wayne Hennessey in the opening 10 minutes.
Mile Jedinak had Palace's first effort on goal, but his header from Yannick Bolasie's corner was saved by David Button.
Adrian Mariappa had to be alert to prevent Palace from going behind in the 21st minute, clearing off the line after Gray's lob had beaten Hennessey.
Bolasie brought a save out of Button 10 minutes later, while Gray shot over from close range as the first half came to a close.
Douglas put the home side ahead three minutes after the interval, and that lead was doubled seven minutes later as Gray fired past replacement goalkeeper Julian Speroni.
However, the visitors hit back immediately, Murray converting from close range, and Chamakh levelled matters after Button had failed to hold on to Bolasie's delivery.
Odubajo's winner meant there was disappointment for Pulis, although he will be buoyed by news that new signing Fraizer Campbell scored a hat-trick in the opening half an hour of a Palace XI's 3-0 victory over Forest Green Rovers.
Brede Hangeland, who signed a one-year deal with Palace on Friday, also started against Forest Green, together with the likes of Barry Bannan and Adlene Guedioura.
