Darlo made a bright start to the game and they went close on two occasions in the opening minutes of the game.
After only two minutes a through ball from Kalusivkako Ngoma found Michael Cummins inside the area, but the midfielder was denied by Swindon keeper Petr Brezovan and then two minutes later Ngoma was involved again when setting up Gaetano Giallanza, whose first time effort was straight at Brezovan.
After a bright opening a mistake on five minutes from Barry Conlon in midfield set Robins forward through Christian Roberts who was up-ended by Patrick Collins at the expense of a free kick 25 yards from goal.
Paul Evans took the free-kick after lengthy treatment for Roberts, and so with seven minutes on the clock the visitors took the lead with a low, curling free-kick from the Welshman that nestled into the far corner of the net.
Conlon seemed desperate to make amends and in the 18th minute he fired narrowly wide after being set up by the impressive Ngoma and then in the 21st minute the visitors again broke to increase their advantage.
Andy Monkhouse beat Darren Holloway down the left flank and his deep cross found the head of Lee Peacock, who headed down for an easy chance for Roberts, which he despatched with ease from six yards out.
Darlington continued to play their passing game and they were rewarded just seconds before the interval when Giallanza was on hand at the far post to sweep the ball in past Brezovan from a deep cross by Craig James.
The visitors were on the back foot in the second period although Peacock was denied in the 66th minute when Darlington keeper Sam Russell finger-tipped the striker's header onto the crossbar.
Darlington introduced new signing Julian Joachim in the 67th minute.
Joachim, who scored two against the Quakers last weekend, almost scored with his first touch after a quick turn and shot from inside the area and then the former Villa striker appealed for a penalty in the dying seconds, but the referee waved play on.