Molyneux opened the scoring in the 71st minute after cutting inside and finding the bottom corner.
Two minutes later he doubled his side's lead when he converted George Miller's cutback from the edge of the area.
The winger’s double was Doncaster’s reward for one of their strongest displays of the season in Wiltshire.
They twice went close to opening the scoring in the first period through Miller and Kyle Hurst and kept the hosts at bay for large periods to secure back-to-back league clean sheets for the first time this season.
Swindon almost grabbed the first goal on the stroke of half time when Jake Cain's free-kick smashed against the crossbar.
That chance was the closest the hosts came to scoring in Jody Morris’ first home game in charge.
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1. Match action
Luke Molyneux celebrates his second goal with George Miller. Photo: Howard Roe/AHPIX LTD
2. Jonathan Mitchell - 7
Tipped a long-range effort from Jonathan Williams over the bar in the first half and collected several dangerous balls into the box. Photo: HOWARD ROE
3. James Maxwell - 8
Like Brown, used the big surface to get forward regularly, particularly in the opening stages. Another solid defensive showing too. Photo: Andrew Roe/AHPIX LTD
4. James Brown - 8
Provided a good outlet down the right-hand side as Doncaster's wing-backs were given freedom to get forward. Almost scored at the death. Photo: HOWARD ROE