The former Bradford City striker powered home a header on the hour mark before stepping up two minutes later to blast his 16th of the season from the spot.
Strike-partner Charlie Austin had fired Swindon into a richly deserved lead, powering past Casper Ankergren in a near-perfect display by the hosts.
Leeds were sluggish throughout and failed too trouble their former shot-stopper David Lucas with more than 3,000 fans making the trip south for a sell-out County Ground.
Austin set the ball rolling, collecting Simon Ferry's pass before placing his finish into the far corner.
Ankergren kept the score down before half-time, denying Jon-Paul McGovern, but Leeds failed to wake from their slumber as they were put to the sword by rampant Swindon.
Lescinel Jean-Francois' swinging cross was finished with style by Paynter before Ankergren was fortunate not to see red as he pulled back Austin inside the penalty area two minutes later.
Paynter duly converted and should have sealed his hat-trick twice inside the last 10 minutes only for Ankergren to spare Leeds further embarrassment.