Midfielder King's 11th goal of the season also saw City stretch their unbeaten League run to 18 games.
City's first attempt on goal arrived on 12 minutes when Lloyd Dyer and Steve Howard combined to set up Matty Fryatt who flashed a 16-yard shot just wide.
The home side continued to dominate and it was 19 minutes before David Martin was called into action when he safely dealt with a tame shot from Kevin Amankwaah.
Petr Brezovan came to Town's rescue on 27 minutes when he blocked King's close-range drive after Steve Howard had knocked down a Matt Oakley corner.
Fryatt was then a fraction from connecting with Tom Cleverley's through ball as City continued to probe.
But it was Swindon who took the surprise lead on 38 minutes when Simon Cox plundered his 19th goal of the season.
He was put through by Billy Paynter and the Swindon striker outstripped City's rear guard to unleash a shot which was blocked by Martin, but Cox followed up to sweep home the loose ball.
After the break Martin hobbled off injured to be replaced by debutante keeper Carl Pentney.
City struggled to discover any rhythm after the re-start but went close on 63 minutes when Michael Morrison crashed a header from an Oakley corner against the underside of the bar.
Youth Acadamy graduate Pentney rescued City on 70 minutes when he stood his ground to block Cox's 15-yard angled drive.
The young keeper again made his mark again on 75 minutes when he instinctively stuck out a leg to keep out a Hal Robson-Kanu shot.
Brezovan kept Swindon in front when he back-pedalled to tip over Fryatt's goal-bound lob before King headed wide with the goal at his mercy.
But King showed great composure on 86 minutes when he stepped inside a defender to steer a shot beyond the keeper from 16 yards.