Cherries shrugged off the absence of star striker Brett Pitman to record their first victory at Tranmere since 1981.
Goals from Josh McQuoid, Michael Symes, a penalty, and Mark Pugh earned Eddie Howe’s men all three points at Prenton Park.
A superb first-half display from Cherries was capped by goals from McQuoid and Symes inside the opening 17 minutes.
McQuoid had been drafted in as a straight swap for Bristol City-bound Pitman as boss Howe made one change to the Cherries starting line-up.
And the 20-year-old took just nine minutes to make his mark when he fired Cherries in front with a well-taken goal following a splendid pass from Symes.
McQuoid outstripped two defenders from just over the halfway line before coolly slotting the ball past advancing Tranmere goalkeeper Joe Collister from just inside the 18-yard box.
Liverpool-raised Symes made it 2-0 from the penalty spot in the 17th minute after he had been tripped by Rovers defender Ian Goodison.
Cherries piled forward in search of a third and McQuoid and Steve Purches were both denied by last-ditch blocks.
Marvin Bartley joined in the shooting fun when his rasping 25-yarder flashed past the upright two minutes before the break.
Shwan Jalal, who was rarely troubled in the first half, got down smartly to smother a speculative shot from Josh Labadie on the stroke of half-time.
Cherries kept up the tempo after the break and Pugh went close to adding a third, only to be denied by a save from Collister.
However, the flying winger made it 3-0 after 61 minutes when he netted following good work by McQuoid.
Cherries: Jalal, Purches, Pearce, Cooper, Wiggins, Feeney (Hollands, 76), Robinson, Bartley, Pugh, Symes (Stockley, 83), McQuoid (Bradbury, 63).
Unused subs: Arter, Cummings, Garry, Partington.
Booked: Purches, McQuoid, Robinson.
Rovers: Collister, Darville, Goodison, Wood, Cresswell, Labadie (Fraughan, 63), Welsh, McLaren (Brown, h-t), Thomas-Moore (McGurk, 68), Showunmi, Gornell.
Unused subs: Cathalina, Blanchard, Bakayogo, Miotto.
Booked: Goodison, Labadie, Thomas-Moore.
Referee: Carl Boyeson (Hull).
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