Head coach Ian Holloway made six changes following the weekend defeat to Shrewsbury Town, and was rewarded with a spirited response on the road.
Barnet took an early lead after nine minutes.
Idris Kanu broke down the left and delivered to the back post, where Mark Shelton headed the ball back across goal. Adam Senior reacted first to fire home from close range.
The hosts continued to threaten, with former Swindon midfielder Nnamdi Ofoborh heavily involved.
He clipped a ball towards Senior at the back post, whose header was cleared by Jamie Knight-Lebel only as far as Ofoborh. Joseph Kizzi was then found inside the area, but could not apply the finish.
Ofoborh later struck the post after Kanu’s deflected cross fell kindly on the edge of the box, as Town endured a difficult spell.
Swindon almost levelled when Kizzi attempted to head back to goalkeeper Cieran Slicker. Clarke intercepted, but Slicker reacted well to deny him.
Junior Hoilett added urgency before the break, slipping Aaron Drinan through at a tight angle. The striker beat two defenders but found only the side netting.
Barnet led 1-0 at the interval.
Swindon responded quickly after the restart.
Jake Batty’s long throw caused problems in the home area. Senior’s attempted clearance dropped into the centre of the box, where Clarke reacted sharply to stab home the equaliser five minutes into the second half.
Momentum swung in Town’s favour.
Hoilett again found space and delivered towards Drinan, whose header did not trouble Slicker. Swindon pressed higher and began to control possession.
The decisive moment came on 63 minutes.
Joe Snowdon kept the ball alive before finding Fletcher Holman near the by-line. Holman cut the ball back to Clarke on the edge of the area, and the skipper calmly slotted into the bottom corner to complete his brace.
Town pushed for a third.
Drinan nodded into Snowdon’s path, who squared across goal for Holman, but the forward was unable to convert. Drinan later broke clear but lost his footing at the crucial moment.
Barnet applied late pressure, but Swindon managed the closing stages professionally. Snowdon was booked in stoppage time as the visitors saw out the contest.
The result sees Swindon return to winning ways ahead of Saturday’s lunchtime visit from Crewe Alexandra.
