Former Reading striker Cureton delivered an unstoppable volley two minutes before the interval to give the Robins their second consecutive home win.
The 1-0 scoreline somewhat flattered Walsall, who were poor throughout and Paul Merson's side now face a real relegation scrap.
Cureton alone could have helped himself to a hat-trick and spurned a number of opportunities throughout the game.
After just ten minutes a deep cross from full-back Jack Smith was somehow headed wide by Cureton as he stood unmarked in the six-yard box.
Nine minutes later the impressive Ricky Shakes outpaced Dan Fox down the Walsall left and crossed for Scottish striker Lee Peacock to head home, only for Andy Oakes to make a last gasp save.
Shakes then outpaced Fox again down the left and delivered a right-foot effort that drifted just wide.
Walsall's only real chance of the half came when Kevin James pounced on a slip by Swindon skipper Sean O'Hanlon and hit a left-foot shot which Rhys Evans saved comfortably, diving to his right. But James perhaps should have fed Mads Timm who was in a much better position.
As the second half progressed Walsall showed no real signs of clawing their way back and Cureton nearly helped himself to another stunning volleyed goal when he turned on his right foot inside the penalty area, latching on to a long ball by Smith, only to see his effort tipped wide by Oakes.
In desperation the Saddlers threw three upfront but still failed to break down a staunch Robins backline.
And the home side nearly doubled their lead on 64 minutes when substitute Charlie Comyn-Platt robbed Michael Leary in midfield.
He fed Aaron Brown who out-ran Craig Pead only to hit a left-foot shot inches wide of Oakes' far post.