A close-range header from central defender Anthony Gardner shortly before halftime gave London club Palace a slender advantage for the second leg in south Wales in two weeks.
The first league Cup semi-final not to feature a top flight club since 2000 was a lively contest and Cardiff, FA Cup runners-up in 2008, will still feel confident of booking another trip to Wembley.
They enjoyed plenty of possession and created the better chances, especially before the break most notably when Ben Turner's header narrowly cleared the bar.
Kenny Miller should have scored for the visitors, roared on by a large travelling contingent, but his low curled shot crept just wide of the post.
Gardner, who was capped for England during his days at Tottenham Hotspur, broke the deadlock in the 43rd minute when he nodded home at the far post after Cardiff keeper Tom Heaton failed to intercept a whipped-in free kick by Darren Ambrose and the ball flicked off the head of Kagisho Dikgacoi.
Palace, who shocked Manchester United in the quarter-finals, were content to play on the counter-attack in the second half as Cardiff searched in vain for an equaliser.
Premier League leaders Manchester City host Liverpool in the first leg of the second semifinal on Wednesday.
