Peter Taylor's men would have guaranteed promotion with a win, but they ultimately had to rely on third-placed Tranmere being held against lowly Blackpool at Prenton Park.
The Tigers could perhaps be forgiven for letting their nerves get the better of them, with a sense of history hanging potently in the air. Even the club's mascot, Roary the Tiger, had been given a brand new head for the occasion.
But Swindon had set out to spoil the party before the game had even kicked off, insisting that the same ball be used throughout, instead of the "multi-ball" system the Tigers are used to.
City started with all the determination one would expect - many commentators predicted four or five goals for the hosts.
In the fourth minute, Junior Lewis was felled just outside the area, but Andy Dawson's subsequent free-kick hit the Swindon wall. Seconds later, Stuart Elliott was just inches away from getting his head to the ball as it flashed across goal, as the Tigers put on all the early pressure.
Lewis had a great chance to score in the 12th minute when he found himself one-on-one with keeper Rhys Evans 20 yards out, but he shot just wide.
Swindon were doing their best to frustrate City, man-marking tightly in their own half. But the Tigers nearly found an opening in the 15th minute through Ryan France, whose quick stab at the ball from 10 yards was well held by Evans following Nick Barmby's clever ball in.
A powerful header from Elliott went just wide three minutes later, after another perfect ball in from Barmby, which was this time a free-kick on the right-hand side of the box.
Evans did extremely well in the 22nd minute to tip over Craig Fagan's acrobatic 18-yard effort, while a few minutes later Tigers captain Ian Ashbee blazed over after being played in by another superb cross-field ball from Barmby.
On the half-hour mark, Dawson hit the side netting following a great counter-attack started by some clever work by Fagan.
Swindon did have a couple of attacking periods, with David Duke working particularly well down the left, but the Tigers kept them at bay confidently.
Evans was tested again in the 37th minute, doing very well to tip over Barmby's powerfully struck free kick from the edge of the area, and Lewis shot wide a minute later.
Rory Fallon got Swindon's first shot on target in the 40th minute, but keeper Bo Myhill held the 20-yard shot comfortably.
Swindon approached the second half with much more menace and tenacity, wrestling back more possession and creating a number of openings, including one by Grant Smith, who should have done better from 12 yards in the 50th minute.
A superb save from Myhill three minutes later, tipping over Fallon's point-blank header, was the only thing preventing Swindon from raining on the Tigers' parade.
Minutes later, Fallon then raced to tackle Myhill, who was about to clear the ball, but the City keeper just managed to get a foot in and divert the ball across goal.
The Tigers' promotion party was on the verge of fizzling out, and the fans, though in good voice, were becoming audibly frustrated.
Brian Howard went on an excellent run in the 60th minute, and was allowed to tee up a powerful shot that went wide.
Evans again denied City in the 65th minute, tipping over Fagan's fluky chip which had been heading just under the bar. But there were a few moments of chaos in the Tigers' box minutes later, as Swindon threw everything at City, who somehow cleared the danger.
Kevin Ellison was kept out as the Tigers hit back in the last 10 minutes, with Evans smothering the danger after panic in the Swindon box, and the referee later ignored the Tigers' pleas for a penalty after Ellison was bundled over, but still Swindon's defenders battled on.
There was unbelievable drama in the dying seconds, as, with the fans in the knowledge that Tranmere were drawing, City were awarded a last-gasp penalty.
With the KC Stadium a cauldron of tension, Fagan stepped up. However, Evans was again equal to the task and beat his effort away - but it didn't matter, as it was full time at last and the celebrations could begin.
The stadium's state-of-the art PA system was inaudible over the deafening crowd, who watched in awe as the team did a lap of honour.