Brighton had the better of the game, particularly in the first half, and most of the limited number of opportunities but they could not find a way past Danny Coyne.
The struggling home side suffered a blow before the match, with skipper and top scorer Nicky Forster ruled out by a knee injury sustained in training.
Brighton had the consolation of welcoming back Glenn Murray following hernia surgery and the former Rochdale marksman gave their attacking efforts more impetus.
They could have scored inside the opening minute when diminutive winger Dean Cox latched onto a Lloyd Owusu header to scissor-kick just wide of the target on the volley.
Murray found the side-netting and also toe-poked over before Cox forced Coyne into a flying save with a 20-yard shot.
Brighton goalkeeper Michels Kuipers, returning from a month long absence with hamstring trouble, had nothing to do until the 25th minute when he made a one-handed stop from Ian Moore's angled shot from eight yards.
Tranmere improved after the break and Kuipers had to block a shot by Charlie Barnett moments after the restart.
Much of the second half was played in torrential rain and hail. Brighton weathered the storm from their high-flying visitors, although Kuipers was called into action again to divert Antony Kay's header from a corner with nine minutes left.
Craig Davies switched to centre-forward from the right wing once Murray limped off in the second half and the Welshman had a chance to snatch the points for Brighton five minutes from time.
Adam Hinshelwood's lay-off found Davies in space but Coyne was smartly off his line to make a smothering save.