Michael McIndoe scored a penalty and missed one as Doncaster, who were 3-0 up at one stage in the game, held on for three precious points.
McIndoe, with six in six from the spot this season, had his 15th-minute penalty saved by Gareth Stewart after Lewis Guy had been impeded.
Paul Heffernan gave Rovers a 41st minute lead when he picked up a loose ball, after Sean Thornton's run, beat a defender and slotted the ball home.
Rovers were coasting to a comfortable win thanks to two goals after the hour.
First McIndoe atoned for his earlier penalty miss by converting from the spot after Steve Foster had been held back by Marcus Browning inside the six-yard box.
Then a flowing move involving five Rovers players culminated in Ricky Ravenhill lashing the ball home after Guy's volley had been headed goalwards by Thornton.
But within a minute, Bournemouth pulled a goal back when Andrew Surman found space in the box and fired past Jan Budtz.
Budtz then dropped a clanger when he fumbled a cross at the feet of Dani Rodrigues who made no mistake in the 77th minute.
Paul Green calmed the nerves in the home camp when he curled home a fourth goal in the 85th minute, but Rovers were still sweating when James Hayter missed a penalty in injury time.