Rovers, now without a win in seven games, defended stubbornly in the first half but Brighton took a stranglehold once they had gone ahead.
The Seagulls broke the deadlock after 63 minutes when defender Ian Goodison handled Elliott Bennett's ball into the box and Wood, on loan from West Brom, stepped up to drill home his eighth goal of the season from the penalty spot to the right of goalkeeper Tony Warner.
Brighton keeper Casper Ankergren came to his side's rescue moments later with an important scrambling save to keep out an Andy Robinson effort. But Murray doubled Brighton's lead after 72 minutes with his 18th goal of the campaign, turning skilfully to fire past a static Warner following a long throw into the area.
Struggling Tranmere had an early penalty appeal turned down when the on-rushing Robinson appeared to be barged off the ball on the edge of the area by Marcos Painter. Brighton midfielder Matt Sparrow was thwarted by Warner after a through ball and Murray was unable to get sufficient purchase on the rebound.
Sparrow had a 25-yard shot saved by Warner after Bennett laid the ball to him midway through the first half. And Bennett struck a low free-kick just wide in first-half stoppage time after Robinson was penalised for obstruction.