The goalkeeper made the vital save after a drab affair, which the defences dominating throughout the game, had disappointingly ended goalless after 90 minutes.
Billy Paynter should have opened the scoring for the home side after seven minutes when John Paul McGovern's driven cross found the Swindon striker in front of goal only, but he hit his effort well wide.
A disputed corner after 15 minutes for Norwich found Michael Nelson, who headed back into the goal area, but Jamie Cureton managed to hit his shot well over from close range.
After 20 minutes a slip by Swindon defender Gordon Greer let in Cody McDonald but Swindon keeper Phil Smith came out to block the shot with his legs.
The game drifted on to the inevitable penalty shoot-out and when Easton's kick was saved, Cureton stepped up to slot the winning penalty home with ease.