Saints took the lead at a rainswept St Mary's when Shane Long struck his first goal since February 2017 in the 17th minute, ending his drought by smashing home Jeremy Pied's low cross from inside the area.
Alex McCarthy made a superb low save to deny Christian Benteke as Palace pushed for an equaliser after the break, and they finally found one in the 69th minute when McArthur drove home from close range after Benteke's flick-on.
Then Milivojevic curled home a sumptuous winner from 20 metres from Andros Townsend's cut-back with 10 minutes remaining to help Palace leapfrog their opponents and move into 14th, while Southampton drop to 17th, above the relegation zone only on goal difference, and departing to boos from their fans.
(Reporting by Rik Sharma,; Editing by Neville Dalton)
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