Kuryu Matsuki, Flynn Downes, Cyle Larin, Ross Stewart and Finn Azaz struck for the Saints, who are now 16 games unbeaten in all competitions.
That run includes Saturday's shock victory over Premier League leaders Arsenal to reach the FA Cup semi-finals.
Automatic promotion still appears out of the Saints' reach after their terrible start to the campaign under Will Still.
But they have been transformed since Tonda Eckert took charge and look like the team to beat should they seal a place in the play-offs.
Southampton move two points ahead of Wrexham and have a game in hand on their rivals for a place in the top six.
Wrexham are aiming for a fourth consecutive promotion after a remarkable rise under the ownership of Hollywood stars Ryan Reynolds and Rob McElhenney.
Despite making six changes from the side that stunned Arsenal at the weekend, Southampton picked up from where they left off in ending the Gunners' hopes of a treble.
The visitors had already hit the crossbar twice through Larin and Azaz before Matsuki was afforded far too much space inside the box to fire into the bottom corner.
Downes had even more time to pick his spot after a free-kick broke the way of the former West Ham midfielder on 22 minutes.
Wrexham had barely laid a glove on their opponents until they got back into the game when Josh Windass slotted home.
Yet, just as the home side were enjoying their best spell in the game after the break, Southampton pounced to seal the points.
Larin took advantage of a loose pass by Lewis O'Brien and sprinted clear on goal to fire into the far corner.
Stewart then replaced the Canadian international and rose highest to head in Welington's free-kick.
Azaz rounded off the scoring from close range on a demoralising night for Wrexham's Premier League dream.