The visitors, under new manager Geraint Williams, put in a dogged performance and fully deserved their point.
There was a very quiet opening as both sides sized each other up. Tranmere's Ed Sonko had an early effort blocked and Antony Kay fired wide from the edge of the box.
Andy Taylor had no better luck with a 20-yarder which cleared the bar by a long way.
Orient with just Ryan Jarvis in attack, took 21 minutes to get in a blow and then loan signing Jimmy Smith brought a comfortable save from Danny Coyne.
Sonko headed wide from a Taylor cross just short of the half hour as Tranmere tried to step up the pace. Then a Kay header from a Chris Shuker corner posed no threat to Jamie Jones.
Orient enjoyed a good spell approaching the break which culminated in a sound save by Coyne from a Jason Demetriou drive which followed good work by Dean Morgan.
Rovers almost gained a breakthrough on 43 minutes, but Jones brilliantly tipped over a fine strike by Sonko.
Sonko did not start the second half for Tranmere with Bas Savage taking over. Kay had a good chance for the Merseysiders from a Shuker corner, but headed off-target when he should have done better.
Demetriou almost capitalised on a Ryan Shotton error, but his angled effort on 52 minutes cleared the far post.
Steve Jennings and Ian Moore failed to trouble Jones as Tranmere stepped up the pace in the last 20 minutes, before Ian Moore volleyed just wide.
Jennings should have broken the deadlock on 86 minutes after clever play by Shuker, but shaved the outside of a post.
Tamika Mkandawire, Demetriou and Jordan Spence were excellent for Orient, Shotton impressed for Rovers.
Tranmere's Ben Chorley and Demetriou for Orient collected the only cautions of a poor game.