Ivory Coast forward Wilfred Zaha, who has repeatedly been linked with a move away from Selhurst Park, was unmarked when he produced a clinical finish from Andros Townsend’s impressive cross.
Nathan Redmond wasted a great chance following a slick Southampton move, while Palace keeper Vicente Guaita produced an excellent save to keep out a volley by Che Adams.
Guaita then produced a reflex save to deny Danny Ings a last-gasp equaliser.
Saints defender Kyle Walker-Peters had a red card downgraded to a yellow in the second half after referee Jonathan Moss reviewed his foul on Tyrick Mitchell on the pitch-side monitor.
Southampton were denied an equaliser with 53 minutes on the clock when Guaita produced a superb point-blank save to deny Adams from close-range.
Palace, with midfielder Cheikhou Kouyate at centre-back alongside Dann, were able to regroup though and the next opportunity for the visitors was 20 minutes later but captain James Ward-Prowse curled a free-kick over.
Zaha had the ball in the net again during the final exchanges, but it was ruled out for offside and Eze had entered proceedings to warm applause from those allowed inside the stadium by this point.
Saints thought they had snatched a point in the fifth minute of stoppage time and yet with Ings ready to celebrate his header, Guaita produced a fine diving save.
Crystal Palace 1 – 0 Southampton Highlights