Rovers made the brighter start and United keeper David Lucas had to flick over a dangerous cross by John Johnson.
But Leeds grabbed the early advantage following a corner conceded by Johnson. Fraser Richardson helped on the cross by Robert Snodgrass for Enoch Showunwi to beat Danny Coyne from eight yards in the 11th minute.
It took Rovers just six minutes to find an equaliser. Lucas came for an Andy Taylor free-kick, but it was Antony Kay who made contact with a header which sailed into the net from 12 yards.
It was real cut and thrust and it needed a fine save by Coyne to deny Luciano Becchio following another Leeds corner on 25 minutes.
There were bookings in quick succession for the Leeds pair of David Prutton and Paul Telfer, both for fouls on Bas Savage.
A Snodgrass blast was just over from 20 yards for United and Edrissa Sonko was high and wide from an angle for Rovers just ahead of the break.
The Merseysiders took the lead three minutes after half-time. Ian Moore netted with a fine shot in off a post from 20 yards when set up by Kay.
Soon afterwards Sonko was only just off target after a corner.
United thought they had levelled on 61 minutes but a Becchio header from a Snodgrass corner was ruled out for a push on Kay. Then Bechio and Jon Douglas had no joy with headed efforts.
Snodgrass was always in the thick of the action for the visitors. Kay and Taylor did well for the home side.
Tranmere's Mark Wilson was stretchered off in the 82nd minute with Gareth Edds taking over for the closing stages.