The visitors put pressure on the hosts early before the Eagles took control and looked the more dangerous side throughout, the Frenchman finally breaking the deadlock just before the break.
Che Adams returned to Ralph Hasenhuttl's starting line-up and at times looked to spoil the Eagles' afternoon, but could only watch his best effort ping off the post. It was a satisfying third consecutive home win for the hosts, who now will not return to Selhurst Park until Boxing Day.
Palace's first opportunity came when Michael Olise curled a free-kick toward goal but saw it tipped away by Southampton keeper Gavin Bazunu. Palace seized momentum as captain Luka Milivojevic's powerful shot was blocked by Lyanco.
The Eagles came close following a fancy piece of footwork by World Cup hopeful Eberechi Eze to feed Wilfried Zaha, whose attempt from the left rolled across the face of goal.
The hosts somehow avoided what has become an unwelcome signature for Patrick Vieira this season, conceding a first-half opener, after Adams wove through multiple Palace shirts only to direct his shot at the right post.
And it was frustration at the other end for Eze as Bazunu blocked a low effort at the left post before he drilled a second attempt from the same spot just wide.
The hosts thought they had the opener when Edouard found the back of the net at the 32-minute mark but the Saints were spared by the offside flag.
Moments later the Southampton keeper did well to save a Joachim Andersen header, but Palace looked close to breaking the deadlock as they held strong in the final third.
It was Edouard who finally found the net, following a few quick touches and putting enough on a neat Tyrick Mitchell pass to slot in the opener from six yards out on 38 minutes.
Southampton looked re-energised to start the second half, though Palace keeper Vicente Guaita was not called into action until he was forced into a kneeling save from a powerful Adams shot then parried away Stuart Armstrong's low effort at the left post.
Guaita blocked another Adams attempt as the Eagles looked to extend their advantage, Zaha latching onto a long ball from Olise but unable to get a shot in and Milivojevic's long-range effort sailing well high of the woodwork. Saints skipper James Ward-Prowse eyed up an equaliser but dragged his shot wide as the game neared the 70th-minute mark.
Substitute Jordan Ayew drove into the area on the right, looking like he might try to double the Eagles' lead only to waste the chance when he elected to pass.
The decision did not prove costly as Palace fended off five minutes of added time and Guaita punched away any hope of the sides splitting the points.
Additional reporting by PA.