Wyke has scored six goals in 10 matches since joining Bradford on January transfer deadline day in a B£250,000 deal from Carlisle.
Swindon, who have now lost four matches in a row, shook Bradford by going in front in the 74th minute. Jonathan Obika broke away down the middle before sending Nicky Ajose through on goal and he slotted the ball into the far corner of the net.
Bradford, who dominated the game and created most of the chances, equalised in the 85th minute when outstanding winger Mark Marshall crossed from the left and Wyke scored with a close-range header.
Wyke then scrambled a winner in the 90th minute. Marshall again crossed from the left and, as substitute Alex Jones slid in with Swindon skipper Nathan Thompson, the ball ran loose for Wyke to force the ball over the line.
Wyke also went closest to scoring in the first half when he headed Marshall's cross against the post in the 14th minute.
Bradford, unbeaten in seven matches, also threatened in the 38th minute when Swindon keeper Lawrence Vigouroux blocked defender Nathaniel Knight-Percival's header on the line from Tony McMahon's free-kick.
Vigouroux saved his side in the 51st minute when he palmed defender Rory McArdle's header away for a corner as Bradford kept up the pressure which eventually paid off.