Three goals in the opening 18 minutes from the hosts, including two from Ben Gladwin and another from Michael Smith, had the Country Ground banking on a Wembley appearance.
But Sheffield United forced their way back into it before half-time thanks to an own goal from Nathan Thompson and Chris Basham’s header.
Smith scored from the penalty spot just before the hour-mark to make United’s task harder again, before Stephen Davies’ bullet header brought the visitors within two goals.
Jonathan Obika’s cool finish made it 5-3 to Swindon on the night and 7-4 on aggregate with six minutes remaining, but it was not the end of the scoring as goals from Matt Done and Che Adams within three minutes of each other put Sheffield United within one goal of levelling the tie.
Swindon held on during a nervy nine minutes of injury time, and will now face Preston at Wembley on May 24 as they look to return to the second tier for the first time in 15 years.
Following on from their last-gasp 2-1 win at Bramall Lane on Thursday, Swindon started on the front foot and were ahead through Gladwin’s superb half-volley from 20 yards on four minutes.
Gladwin had a simpler task six minutes later to make it 2-0, converting at the far post after Nathan Byrne won a 50-50 before finding the midfielder unmarked.
And it was 3-0 within 20 minutes as a hopeful cross found its way to Smith via the shins of Kieron Freeman, the striker taking a couple of touches before slotting into the bottom right corner.
But the visitors got a goal back a minute later as Jamie Murphy’s low cross deflected off Thompson to wrong-foot goalkeeper Wes Foderingham.
And seven minutes before half-time Basham got in on the act for his first Blades goal, heading in from 10 yards out from Bob Harris’ perfect cross.
The goal glut continued in the second half, Smith regaining Swindon’s three-goal lead on aggregate from the penalty spot after goalkeeper Mark Howard had hauled down Jermaine Hylton.
Substitute Davies replied just moments later with a far-post header from Harris’ cross, before Obika, another substitute, cut inside and stroked home a classy finish to give Swindon a three-goal lead in the tie yet again.
But Sheffield United were not finished, and scored twice in three minutes to create a tense finish. First, Done slammed home from close range after Foderingham had kept the Blades out with three fine saves, then substitute Adams curled in a fine effort from just inside the area.
The Yorkshiremen could not force one more chance in the lengthy added time, cue a pitch invasion from the home supporters after witnessing the highest-scoring game in play-off history.