In the first league fixture between the two sides for 22 years, Swansea calved out an early opening when in-form striker Vipotnik fashioned half a yard of space on the edge of the box, only for his shot to be blocked in the nick of time.
Soon growing into the game, Wrexham took the lead after 14 minutes when George Dobson’s excellent diagonal pass found the onrushing Ryan Longman, whose left-footed cross was diverted into his own net by Swansea’s Burgess. It was the third time the Australian had put through his own net this season.
Huffing and puffing in search of an equaliser, Swansea had a penalty shout waved away when Ronald Pereira claimed he was fouled by Wrexham’s Callum Doyle, before Burgess - looking to make amends for his own goal - stabbed an effort over the bar from close range.
Picking up from where they left off at the end of the first half, Swansea were lively going forward, with captain Ben Cabango almost getting on the end of a teasing cross which brought about an important clearance from Wrexham’s Longman.
Swansea defender Burgess was soon breathing a huge sigh of relief when he gave the ball straight to Ben Sheaf, who was able to pick out Josh Windass. However, the forward - rather than look to pick out Kieffer Moore at the back post - opted to shoot instead, failing to hit the target from inside the box.
Dominating possession as time ticked away in the second half, Swansea came close to finding that elusive leveller when midfielder Marko Stamenic curled a lovely effort just wide of the far post.
However, with Wrexham continuing to drop deeper and deeper, the Jacks soon had a deserved equaliser when Vipotnik turned brilliantly inside the penalty area before seeing his effort deflect in off Dominic Hyam.
And in the dying embers of the game, Swansea were gifted all three points when Wrexham goalkeeper Okonkwo dropped the ball right into the path of substitute Idah, who had the easy task of tapping into an empty net.
After securing a third win in their last five league games, Swansea moved up to 17th in the Championship table, with Matos' men now just two points behind 15th-placed Wrexham. Conversely, Parkinson's side are now winless in their last five league fixtures, drawing three and losing two.