The faithful, who had followed the Vauxhall Conference side from Hertfordshire into the depths of Wiltshire, poured on to the pitch at the final whistle to congratulate their heroes who had mastered their opponents - and the elements - to record a victory that seemed no more than a distant dream after only five minutes.
"We coped with the conditions much better than Swindon did and I thought that we were the better team and played the better football and as a result deserved everything we got,'' the Stevenage manager, Paul Fairclough, said.
"We were optimistic about our chances before the game and even though we conceded an early goal I felt we were always in command of the game and it's a brilliant result for us."
Guiliano Grazioli, Stevenage's match-winner who only joined the club from Peterborough two weeks ago, said: "This has been one of the best experiences of my life. To score the winning goal and knock out a Football League club is like a dream come true."
Swindon had survived a second-minute fright when Fraser Digby's goal-kick went straight to an opponent's feet and the keeper had to save face with a brave stop at the feet of Grazioli. It was something of an omen, though, as Digby's discomfort when facing the gale-force wind that raged across the County Ground in the second half was to decide the game's outcome.
It looked as though the difference in class had been restored as quickly as the fifth minute when Mark Walters cracked a fierce 25-yard right- foot volley beyond Des Gallagher.
But far from being intimidated, the non-leaguers moved purposely towards a deserved equaliser in the 23rd minute. Full-back Michael Love's deep cross caused enough confusion in the home defence for Jason Soloman to pounce on the loose clearance and take deliberate aim from 12 yards to curl his left-foot shot beyond Digby's reach.
The pouring rain and howling wind had made life difficult in the first half but as it reached severe gale force after the interval it was Stevenage who mastered the elements.
Swindon had re-started with a flurry of determination to show they were the superiors, but Gallagher was equal to all that came his way. At the other end Digby's discomfort, when every time he cleared the ball the wind hurled it back at him, was growing. On one occasion the keeper was only yards short of making football history by conceding a corner from his own goal-kick.
But it was the tempest which had the last word. In the 65th minute another Digby goal-kick was blowing in the wind and Gary Crawshaw gathered the ball when it dropped, jabbed a short pass to Grazioli who poked a shot low into the corner.
For a moment Swindon thought they had avoided disgrace when Mark Robinson got the ball into the net but it was ruled offside. That recovery was beyond them was underlined when four minutes from time Lee Collins hammered a 30-yarder just under the bar only to see Gallagher tip over.
Steve McMahon, the Swindon manager, said: "None of our players can escape criticism for this because no one played well. There are no excuses, the weather was the same for both sides but we didn't use any common sense and we kept keeping the ball in the air even though the wind was causing us problems. They were schoolboy errors."Reuse content