With the first half drawing to a close, recent signing Cleveland Taylor picked up a loose ball on the right wing after Gareth Roberts had failed to clear for Rovers. Taking just a single touch to control the ball the ex-Scunthorpe winger curled a perfect cross for Graham to smack in his 12th goal of the season past Neil Sullivan.
The blustery conditions should have left the game as a long ball marathon, but both sides proved good to their league positions in a thrilling end to end match. Debutant Grant Smith almost opened the scoring for Carlisle with a sweet volley, before captain Danny Livesey's clever header forced a great full stretch save from Sullivan.
But the visitors had their own chances too and striker Mark McCammon should have done a lot better than head wide after getting in front of home keeper Keiren Westwood from a dangerous cross. Going in 1-0 down at the break forced Doncaster on to the front foot in the second half and Paul Green was unlucky to see his low drive fly inches past Westwood's post.
Graham should have doubled the lead on 53 minutes following good work from Evan Horwood, but his tame six-yard poke was easily gathered by Sullivan. As the game wore on and the Carlisle defence drifted deeper and deeper, Rovers created chances a plenty but none clear enough for an equaliser.
With ten minutes to play Doncaster sub Paul Heffernan played a clever one-two with Richie Wellens inside the Carlisle penalty box. But with Heffernan about to pull the trigger, full-back David Raven pulled out a stunning last-ditch tackle to save the day and ensure that United took all three points.