The hosts were rewarded for a dominant start when Raphael Rossi Branco lashed home Chelsea loanee John Swift's whipped cross on 14 minutes.
Swindon had the chance to double their lead from the penalty spot when Swift was brought to ground in the penalty area by Paul Digby, but Andy Williams saw his spot-kick saved by Ross Turnbull.
Turnbull went from hero to villain after the break, gifting the hosts a second goal six minutes from time by allowing Ben Gladwin's shot from the edge of the area to squirm under his body.
MK Dons started the day in second, but slipped a place to third after their eagerly awaited top-of-the-table clash with Bristol City ended in a 0-0 draw.
Elsewhere, new Gillingham boss Justin Edinburgh - who was appointed earlier on Saturday - was treated to late drama as his new side struck twice in the closing stages to down Sheffield United 2-0 at Priestfield.
A drab game came to life four minutes from time when Jermaine McGlashan lashed home from the edge of the area, before Cody McDonald sealed the win deep into stoppage time, easing his side's relegation worries and putting a dent into Sheffield United's play-off ambitions.
There were no such problems for fellow play-off candidates Preston North End, who returned to winning ways in the league after a surprise defeat at Crawley Town last weekend with a 1-0 victory at home to relegation-threatened Coventry City,
Chesterfield and Oldham - who are fifth and sixth - won 1-0 on their respective visits to Notts County and Scunthorpe United.
At the other end of the table, Yeovil Town won the day's relegation six-pointer with a 2-1 victory over Crawley to leave them three points from safety, while Crewe Alexandra moved four points clear of the drop with a thrilling 3-2 win away to fellow strugglers Colchester United, who slip into the bottom four.
Rock-bottom Leyton Orient's contest with Rochdale was postponed due to a frozen pitch, while the day's other games saw Fleetwood Town held to a 1-1 draw by Peterborough United, play-off chasing Bradford City conceded an injury-time goal in a 2-2 draw at Port Vale, and Walsall beat Doncaster Rovers 2-0 on their travels.
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