Paynter, sent off for violent conduct at Crewe last month, bagged his first league goal since October as the Robins re-ignited their League One play-off hopes.
Simon Cox earlier had marked his full home debut with a goal after some schoolboy defending from the away side, before substitute Christian Roberts made sure of the points late on.
In a frantic opening 15 minutes both sides went close as Cheltenham nearly forced Swindon midfielder Lee Peacock into scoring an embarrassing own goal.
A corner could not be cleared and as the ball ricocheted around inside the penalty area it struck Peacock unawares on the chest before crashing off the upright.
A lively exchange between Cox and Paynter then threatened Cheltenham's goal as the former Reading forward struck a low drive from just inside the penalty area forcing Shane Higgs into a low stop.
It was only a matter of time until the deadlock was broken and unfortunately it arrived in comedy fashion as both Cheltenham's Alex Russell and Gavin Caines conspired to give Swindon the lead.
Russell lost the ball 30 yards out to Paynter and his cross was mis-controlled by defender Caines.
That left Cox almost embarrassed to stab home his first goal since making his move from Reading permanent in the January transfer window.
Further misery for the away side was to come just before the break as the lively Anthony McNamee dazzled on the left-hand side before crossing for Paynter to head his eighth of the season.
The second half failed to match the intensity of the first 45 minutes, leaving Cheltenham boss Keith Downing no choice but to throw caution to the wind and go with three up front.
That bold move left huge spaces at the back and substitute Roberts exploited that within two minutes of coming off the bench by rounding Andy Gallinagh and driving low into the bottom left-hand corner.