Promotion chasing Bournemouth had the better of the opening half, with John Achterberg having to produce excellent saves to keep out headers by Shaun Maher and Steve Fletcher.
Paul Hall missed the target for Tranmere after an excellent run through the visitors' defence, before the Merseysiders got the breakthrough in first-half injury time.
Iain Hume headed his eighth goal of the season at the far post from a left-wing cross by David Beresford.
Rovers had to thank Achterberg for another good stop when he beat out an angled drive by the impressive Wade Elliott on 52 minutes.
There was a scare for Rovers on 70 minutes when substitute Ian Goodison almost put through his own goal and then Achterberg produced a finger-tip save to deny Elliott in nine minutes later.
Hulme almost grabbed a second for Tranmere in the 82nd minute when he saw a 20-yarder deflected on to a post but Hall settled it in the 89th minute with an angled 12-yard effort into the roof of the net as Bournemouth failed to clear their lines.
Tranmere were not helped by first-half injuries to skipper Mike Jackson and Mark Rankine but their replacements, Goodison and Danny Harrison, did well enough.
However, it was Hume who was the star man for the victorious home side.