The hosts quickly forged a two-goal advantage with strikes inside the first 20 minutes from Michael McIndoe (penalty) and Paul Green.
The South Yorkshire outfit appeared to be cruising to victory, but Andy Robinson brought the contest back to life with a reply for The Swans 16 minutes from time.
Donny held firm, however, and put a gloss on the scoreline when Greg Blundell notched his seventh of the season in the dying moments.
Elsewhere, Cambridge United kept in touch with the play-off places with a 2-1 success at Northampton Town's Sixfields Stadium.
The U's led from Luke Guttridge's penalty just before the half hour mark, but were pegged back by Derek Asamoah (pictured) 18 minutes from time.
However, just as the sides seemed set for a share of the spoils, Adam Tann popped up two minutes from time to condemn The Cobblers to their fifth home reverse of the campaign.
Match summaries:
Doncaster 3-1 Swansea
Northampton 1-2 Cambridge
