Leon Best scored the only goal in the 22nd minute but Crewe made O'Driscoll sweat before he was able to finish his Bournemouth career with a victory.
O'Driscoll and his assistant Richard O'Kelly leave for Doncaster next week but not before helping the Cherries to their third successive home win.
Best netted soon after Steven Foley-Sheridan had struck the crossbar with a left-footed drive. Stephen Purches centred from the left and Best got away from his marker to score with a low header.
Bournemouth failed to build on their strong start and Crewe were marginally the better of two sides committed to playing football.
Michael O'Connor, Luke Varney, Jon Otsemobor and Rodney Jack all should have done better with time and space available to them.
David Vaughan saw a low shot well saved by Gareth Stewart but Crewe couldn't have any complaints about the outcome because they struggled in front of goal.
Bournemouth introduced former England winger Darren Anderton as a new signing on the pitch at half-time to signal the start of a new era at Dean Court.
But Anderton, who has been without a club since May when he left Wolves, is still some way short of full fitness and will not be available for at least two weeks.
Bournemouth had match-winner Best and rugged defender Eddie Howe booked for fouls in the second half as Crewe's pressure grew more intense.
Crewe's best chance to equalise fell to West Indian Rodney Jack who flashed a second-half shot over the bar with only Stewart to beat.