Three second-half goals, the first an own goal by Ben Gibson, plus Harry Arter and Simon Francis made it an afternoon to remember for Cherries boss Howe.
Tranmere striker Jake Jervis gave them the lead in the 30th minute when he profited from Bournemouth captain Miles Addison's mistake in the penalty area to turn home a centre from eight yards out.
The game turned in the 55th minute when Marc Pugh's free-kick was flicked goalwards with the last touch adjudged to have come off Gibson and it was later confirmed as an own goal.
Arter drifted past one challenge on the edge of the box soon after and unleashed a rocket into the top left-hand corner of the net to give Bournemouth the lead.
And Howe's side made it three when a well-worked free-kick was passed to Francis and the right-back found the bottom corner from 30 yards out.