Tranmere Rovers defender Ian Goodison gave up the chance to play for Jamaica this week and within 40 seconds of the start had scored an own goal.
Bournemouth's lively winger Max-Alain Gradel centred from the right and Goodison was at full stretch when he turned the ball past his own goalkeeper.
Tranmere hardly looked like a side pressing for promotion and they did not begin to pose questions until after half-time when manager Ronnie Moore made changes.
Bournemouth went further ahead in the 34th minute when Danny Hollands took a corner on the right and, as Tranmere failed to clear, Jo Kuffour turned the ball over the line for his tenth of the season.
Tranmere's Steve Jennings raced clear in the 49th minute but shot wastefully over the bar, but it was a rare opening for lethargic Rovers.
Instead Tranmere owed a big debt to their goalkeeper Danny Coyne, who made a series of outstanding saves in the second half.
Coyne twice denied Kuffour, dived to his right to save a Hollands free-kick and reserved his best for the 58th minute when he saved on his line a header from Jason Pearce.
Tranmere stepped up the pressure near the end as Bournemouth's young side began to tire.
With three minutes remaining they were thrown a lifeline when substitute Antony Kay headed in a free kick from Paul McLaren.
In contrast, Bournemouth's on-loan goalkeeper David Forde had little to do.
But on the hour he was pressed into action for the first time with a great save to push away a close-range shot from Ian Moore.
Tranmere boss Moore made three changes, sending on Calvin Zola, Kay and Craig Curran but Bournemouth held out grimly for a surprise win.
Bournemouth manager Kevin Bond was forced into making numerous changes and he gave full debuts to Josh McQuoid and Billy Franks, who played a full part in Rovers' shock defeat.