SLIMBRIDGE picked up three points after a run of five games without a win on a heavy pitch at Yarnton Road.
The first half ended even with Dan Holloway's penalty for the visitors cancelled out by Jimmy Deabill's equaliser.
But a Marvyn Roberts goal proved decisive in the second half and the Swans returned to Gloucestershire with all three points.
The game ended in controversial fashion, with Kidlington having a last-gasp equaliser ruled out by the final whistle just prior to the ball beating Lee Martin in the Slimbridge goal.
Slimbridge made a good start to the game and nearly went ahead with four minutes on the clock when Dominique Njoya just missed the target.
Sean Bennett went close on the quarter-hour with a shot that Kidlington keeper Matt Dixon fumbled past the post.
The best early chance for Kidlington came on 17 minutes when Nathan Geary's cross was missed by Deabill with the goal at his mercy six yards out.
The Swans took the lead on 19 minutes when Holloway's quick turn saw him brought down in the Kidlington box. Holloway picked himself up to convert the spot-kick himself to give the visitors the advantage.
Jamie Martin then forced a good save from Dixon before his follow-up shot was cleared off the Kidlington line.
Kidlington began to take control of the game and Deabill's overhead kick hit the crossbar before Martin had to make a good save from Tom Boxer's follow-up header.
An equaliser seemed inevitable, and it came on 39 minutes when a corner fell to Deabill in the Slimbridge box and he buried it past Martin.
The second period saw an improved performance from the Swans, and substitute Alex Higgs was crucial in what turned out to be the winning goal.
His run down the right flank saw him deliver a superb cross to the incoming Roberts, and the Slimbridge captain slammed the ball past Dixon to restore the advantage.
The last 20 minutes saw almost continuous home pressure as Kidlington pressed for the equaliser, but Slimbridge – helped by some impressive saves from Lee Martin – held firm.
On 75 minutes Jack Dunmall's mazy run saw his shot drift just wide, and then Jack Johnson brought a great save from Martin with a 20-yard effort.
Two minutes from time Holloway rattled the post with a good effort after neat interplay between himself and Njoya.
Seven minutes of stoppage time saw the Slimbridge goal under siege, and Martin made a crucial stop from Johnson before Deabill hit another pile-driver just wide.
As another Kidlington attack developed the final whistle was blown just as another shot came in on the Slimbridge goal.
The ball beat Martin and Kidlington felt that they had achieved their equaliser, but the 'goal' was ruled out, prompting angry scenes from the home players and officials.
Slimbridge: L Martin, Wellon, Njoya, Halford (Sterling 7), Speers, Bennett, Harding, Ward, Roberts, J Martin (Higgs 65), Holloway.