Matthew Bound, the Swan-sea defender, was the hero with a strike one minute before half-time, but the Swans were indebted to their goalkeeper Roger Freestone who, rushing from his line, made vital clearances either side of the break.
Brian Laws' Scunthorpe came out for the second half in determined fashion but failed to capitalise on their territorial dominance. Alex Calvo-Garcia came closest two minutes from full-time when his header smacked against the Swansea goalpost.
John Hollins, the Swansea manager, said: "This was a terrific game of soccer. I felt we looked as if we could score every time we came forward."
In the other play-off match Rotherham will be the more satisfied after holding Leyton Orient to a goalless draw at Brisbane Road in the first leg.