Rovers moved up to fourth in the National League after they beat Solihull at the Automated Technology Group Stadium.
Two goals from Jake Kirby were added to by an injury time third from James Norwood as Tranmere ran out winners.
Solihull goalkeeper Dan Lewis did well to hold Rovers at bay for the first 20 minutes, twice denying Lois Maynard from close range and comfortably gathering Lee Vaughan's half-volley in the 19th minute.
The former Kidderminster keeper was finally thwarted in controversial fashion in the 25th minute as Jake Kirby rounded him to finish into an empty net.
Rovers celebrate Jake Kirby's contrversial opener for the visitors
The assistant raised his flag for offside, but it was ignored by the referee who waved play on and while the home defence stopped, Kirby took advantage and rounded the goalkeeper to score.
Kirby added gloss to the scoreline with eight minutes remaining, tapping home on the rebound after Maynard's shot was saved, and James Norwood added a third in stoppage time.
Manager Micky Mellon said: "We were made to work very hard for our victory but we expected that - they're a very well-organised team with a good manager.
"We knew we'd have to be all over the pitch. We scored some good goals and have kept a clean sheet so were pleased."
Jake Kirby impressed, as a part of a good all-round performance
Referring to Kirby's disputed goal, he said: "You can't be offside from a goal kick.
"He stayed alert and alive. He's been taught the right way - you don't stop playing until the referee tells you - and we benefited today from that knowledge."
Jake Kirby is congraulated on his second goal of the afternoon, as Rovers take full control of the game
Rovers will host North Ferriby Utd at Prenton Park on Tuesday night.
The side, who hail from the East Riding of Yorkshire, were promoted into the league in 2016 having won the National League North play-offs.