Town's Boxing Day ends in defeat as an early goal is enough for Wrexham to seal a single goal win on Tuesday afternoon.
Wrexham's James McClean scored the game's only goal of the game with a well-taken finish in the 13th minute.
Town brought in Lewis Ward in place of Murphy Mahoney who suffered an injury last time out against Tranmere, with defender Tom Brewitt also returning to the matchday squad for the first time since his injury.
It was a promising start for Swindon, with a succession of corners looking to cause a problem to the Wrexham back-line, but unfortunately, the Red Dragons were able to deal with the early pressure from Town.
A good spell of pressure from Town at the start of the game as Town won a succession of corners that were eventually dealt with by Wrexham.
However, Town's positive start wouldn't last for long though as the Welsh side would find a breakthrough as James McClean would score after moving the ball onto his left foot and putting it into the bottom left.
Swindon would try to make a reply with Dan Kemp trying his luck from range but unfortunately would only find the palms of Onkonwo in goal for Wrexham a minute after the opener.
The game calmed down for the next 15 minutes, with both sides looking to carve out the next clear-cut opportunity. Kemp looked to find a team-mate from a set piece but it whistled past the far post, as Swindon looked to kick on and push for an equaliser.
Both sides looked for the next goal in the remainder of the first half, with Wrexham applying more sustained pressure as Swindon looked to respond in the second 45 very soon.
Half-time: Swindon Town 0 Wrexham AFC 1
The second half started in a much more open fashion, with both sides creating early opportunities, as Jake Young saw an effort blocked from around the outside of the Wrexham penalty area before Lewis Ward produced an excellent save to deny Elliot Lee three minutes after the restart.
10 minutes later, Tyrese Shade tried his luck from some distance and saw his drive go wide of the near post with just over 30 minutes left on the clock.
Ward was called into action for Swindon shortly after, with James McClean denied from close range as the Town keeper managed to get a hand onto the Irishman's effort.
Approaching the 70th minute, Swindon began to turn up the pressure even further, with Jake Young and Williams Kokolo almost able to test Wrexham's Arthur Okonkwo before Town were boosted by the quickfire yellow cards for the away side's James Jones seeing the midfielder given his marching orders with just under 20 minutes left to play.
This is where the game really began to liven up, with Jake Young having the first of these opportunities, that saw the forward in space in the penalty area and could unfortunately only guide his effort over the bar from around 12 yards out.
Only five minutes later another chance would come along when Jake Young picked the ball up deep on the pitch and made a brilliant run dragging defenders out of the position to then lay it off to Saidou Khan, whose first-time curling effort would go inches wide of the far post.
In the dying embers of the game, Jake Young would have another chance at goal as his headed effort would come off the bar and then the rebounded effort from Liam Kinsella would go just wide as the final whistle would shortly follow confirming Town's defeat.
Town's full focus will turn to their game against Forest Green Rovers in three days time, as they look to end the year on a high note.
Swindon Town: L.Ward, Hutton, McEachran (Brewitt 32), Young, Khan, Kemp, Kokolo (Cain 76), Kinsella, Shade, Minturn, Austin (C)
Subs not used: Evans, Genesini, Dworzak, McGregor, Hart
Wrexham AFC: Okonkwo, Tozer (C), Forde (Young 76), Dalby (Fletcher 69), Mendy, Cannon, O'Connor, McClean, Jones, Cleworth, Lee (Davies 89)
Goal: McClean 13
Red Card: Jones 76
Subs not used: Howard, Mullin, Boyle, Barnett
Referee: Tom Reeves
Attendance: 12,271 (1,435 away fans)
