Bournemouth manager Sean O'Driscoll saluted match winner Wade Elliott after his wonder goal which booked his side's place in the FA Cup second round.
Elliott produced the decisive strike eight minutes from half time when he cut in from the right, skipped past the challenge of two Rovers players before curling a 25-yard left footed shot past goalkeeper Kevin Miller.
It was his third goal of the season and his effort extended Bournemouth's unbeaten run to 8 matches.
O'Driscoll said: "Wade is a frustrating player at times but he scored a good goal and that is exactly what he is capable of. It would be good if he could do it every week." "It was a scrappy game in scrappy conditions but my players were very professional with their attitude and we're now hopeful of getting through to the next round and getting a big team in the third round because of our financial problems have been well documented." Elliott should have extended Bournemouth's advantage three minutes into the second half when he thundered a right-foot shot against the crossbar when left unmarked at the far post.
Elliott was also involved in the incident that led to Rovers defender Kevin Austin being sent off in the 72nd minute.
Austin was dismissed by referee Joe Ross after being adjudged to have hauled back Elliott on the line as he attempted to close in on goal.
But Rovers produced a spirited display and wasted two great chances to force a replay.
Bournemouth keeper Neil Moss did brilliantly to deny Lewis Haldane in the 62nd minute when he was sent through on goal thanks to defender Lewis Buxton's sloppy pass.
But veteran striker Andy Rammell produced the miss of the match two minutes from time when he wasted a clear cut scoring chance that would have earned the west country side a draw.
Rammell was unmarked when he received Ryan Willams' corner at the far post but somehow contrived to head over the bar from just 4 yards out.