David McGoldrick's second-half strike sealed a vital victory for Southampton and kept their hopes of Championship survival alive.
McGoldrick found the top corner of the net with a superb strike from outside of the area to secure the points.
Palace keeper Julian Speroni had kept Southampton at bay denying Bradley Wright-Phillips on several occasions.
Late on Shefki Kuqi's last-gasp effort was blocked on the line as Southampton held on for the three points.
Southampton boss Mark Wotte:
"The boys showed 100% commitment and are fighting for their lives.
"The players do not deserve to go down and nor do the supporters.
"We are two points from safety with three games to go and we still have Nottingham Forest to play so it is in our own hands. Three wins will keep us up."
Crystal Palace boss Neil Warnock:
"We could have been playing all night and not scored. We did not have enough quality.
"The fans know how hard I work to get the points, and I think we have done really well with the squad we have.
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