The League One basement club were not without chances of their own against a Rovers side seeking a third win in seven days.
But a first of the season from Steve Jennings shortly after half-time followed by a solo Chris Shuker effort was enough to take the Merseysiders into the top six in the table.
Crewe could have been ahead in the third minute but a Shaun Miller effort was deflected for a corner.
Later Danny Woodards saw a shot blocked before Steve Schumacher fired wide from a great opening on 17 minutes.
It needed a fine save from Danny Coyne to deny Schumacher in the 39th minute when set up by Michael O'Connor.
Antony Kay and Ryan Shotton missed the target from long range for Rovers and Bas Savage made a hash of things in front of a gaping goal.
Then Ian Moore was inches wide from the edge of the box before planting another effort into the side-netting.
Crewe's Byron Moore needlessly gave away possession to hand Tranmere the initiative on 51 minutes and Shuker's cross was netted from close range by Jennings.
It was nearly 2-0 on 65 minutes but keeper Steve Collis did well to block a close-range Moore shot following a pass from Andy Taylor.
But ten minutes later Shuker collected an Ian Goodison clearance near the halfway line and raced through to put an angled shot past the keeper.
Crewe's misery was completed when Billy Jones was harshly red carded in the 87th minute for a challenge on Savage.