The visitors failed to bring a single save from home keeper Dino Seremet throughout the 90 minutes.
The Cherries had luck on their side on more than one occasion and keeper Gareth Stewart made several excellent stops. He was at his best in the opening period keeping out efforts by Jason McAteer and Chris Greenacre.
Danny Harrison wasted a great chance when in the clear midway through the half and Ian Sharps was unlucky to head on to the bar following a 32nd minute corner by Gareth Roberts.
Brian Stock for Bournemouth was off-target with a speculative volley from 45 yards and approaching half time Shaun Cooper headed over following a free kick.
It was much the same after the break with Tranmere always looking the more likely to win. But Calvin Zola had a shot blocked and Sam Aiston, after a good run through, brought a diving save from Stewart.
Greenacre fired just wide from 30 yards and then in the 84th minute Marcus Browning came to the Cherries rescue as he kicked off the line from an Ian Goodison header following a Roberts corner.