The Dragons, co-owned by actors Ryan Reynolds and Rob McElhenney, are aiming for a fourth consecutive promotion to continue their remarkable rise through the English football pyramid.
Victory over Swansea, who have a famous co-owner of their own in musician Snoop Dogg, departed Wrexham sixth and in a play-off place on 60 points ahead of the bulk of this weekend's second-tier Championship fixtures.
Nathan Broadhead opened the scoring in the 25th minute with a low-angled shot, and it seemed that might be the only goal of the game at the Racecourse.
But Wrexham put the result beyond doubt two minutes from time when George Thomason's cross found Callum Doyle, whose downward header took a big deflection off Swansea and Wales forward Liam Cullen for an own-goal.
Wrexham are now six points in front of Southampton in seventh spot, albeit the Saints have a game in hand.
Championship leaders Coventry (77 points) are currently well-clear of second-placed Middlesbrough (69).