Denis Smith's team maintained their unbeaten record but it could have all been over by half-time.
The Robins took the lead in the 28th minute with a stunning 35-yard free-kick from Paul Evans, which keeper Mike Ingham misjudged, expecting a cross rather than a shot.
Just before half-time Wrexham were lucky when normally reliable marksman Christian Roberts missed a glaring chance. Shortly afterwards he put the ball in the back of the net, only to be ruled offside.
Wrexham, who had not played with their normal free-flowing style in the first half, suddenly changed in the second period, causing the Robins problems galore.
Aaron Brown kicked a shot from Mark Jones off his line and Chris Llewellyn forced goalkeeper Petr Brezovan into save.
In the 68th minute Wrexham equalised through Ryan Valentine's penalty after the goalkeeper had bundled over Llewellyn.
Two minutes later Llewellyn put young Mark Jones through to shoot under the keeper and give Wrexham a memorable victory, as Paul Ince and Andy Caton missed chances to equalise in a late rally.