Paynter pounced either side of half-time to take his goal tally against Leeds this season to four, and strike partner Charlie Austin headed Swindon's third in the 55th minute.
Leeds have now gone an alarming 385 minutes without scoring. Their best chance fell to Andrew Hughes in the 29th minute, but he shot straight at David Lucas from six yards before firing the rebound into the body of Lescinel Jean-Francois.
The unfortunate Hughes was at fault for Swindon's opener two minutes before the break as his slip gave Paynter the chance to fire into the roof of the net from the corner of the six-yard box.
There was nothing Leeds keeper Shane Higgs could do to stop Paynter's second goal after 48 minutes. With his back to goal, the striker turned and shot into the top-right corner from just outside the penalty box.
Seven minutes later Paynter turned provider, crossing for Austin to head Swindon's third and leave Leeds in danger of competing in the play-offs.