The 3-0 win was only the second victory on the road this season for the reigning English Premier League champion but came at a cost as Belgium international Eden Hazard limped off in the first half.
Hazard was Footballer of the Year in England last season, but has gone 30 club games without scoring in Chelsea's troubled start to the current campaign.
The 24-year-old was seen to be struggling after attempting a shot and was soon substituted, Hiddink revealing after the match he would be undergoing an MRI scan on a groin injury.
Hiddink, who replaced the sacked Jose Mourinho last month, had previously overseen two draws although Chelsea did beat Sunderland 3-1 just before he was officially appointed.
Those signs of improvement were underlined by the sort of performance which marked Chelsea's title charge last season under Mourinho.
Hazard's departure did not disrupt the visitors and fine passing by Cesc Fabregas and Costa left Oscar with a comfortable opening to side foot home after 29 minutes.
Palace, so impressive this season, was unable to respond although Wilfried Zaha should have done better with a chance which Thibaut Courtois saved.
Soon Chelsea was two-up as Oscar turned provider for fellow Brazilian Willian to blast an unstoppable shot past Wayne Hennessey.
Willian, one of the few shining lights for Chelsea this season, set up the third, his right wing cross spilled by Hennessey into the path of Costa to fire home from close range.
"We know there is a lot of quality in the squad, not just in the starting 11. When they get the ball they can play," said Hiddink, who lost just one match in 23 games when he took charge of Chelsea on a temporary basis in 2009.
The win lifted Chelsea up to 14th, but 13 points adrift of the final Champions League place held by Tottenham Hotspur, which was held 1-1 by Everton in Sunday's late kick off.
Aaron Lennon volleyed Everton ahead against the run of play against his old club but exciting young midfielder Dele Alli leveled for Tottenham just before the break.
Tottenham had the majority of possession throughout and Harry Kane and Ben Davies hit the woodwork with first half efforts, but had to settle for a point to be six points adrift of leader and North London rival Arsenal going into the New Year.