Rovers were without Jay Harris and Gary Taylor Fletcher due to ankle and calf injuries respectively as well as goalkeeper Iain Turner, who picked up concussion earlier in the week. Young academy keeper Patrik Rutter was included on the bench in Turner’s absence.
Wrexham struck the first blow as they went in front on four minutes following a fine run by Adriano Moke who picked out Wes York at the far post, who slotted calmly past Scott Davies.
Shortly after, Adam Mekki nearly found a Tranmere equaliser only for Wrexham goalkeeper Rhys Taylor to make an excellent double save.
Midway through the first half, a Ben Tomlinson cross found the head of Lois Maynard whose effort was well saved by Taylor. Taylor later made another fine save to dive full-length and push a Mekki 25-yard strike to safety.
Tranmere had another chance to equalise only for Mekki’s 18-yard effort to be blocked by the on-rushing Emmanuel Smith.
On the stroke of half time, Vose made it 2-0 to the visitors after a powerful drive from 15 yards found the left corner of Davies’ net.
Deep into first half added time, James Norwood pulled a goal back for Rovers, firing a powerful drive past Taylor into the roof of the net, following good work by Maynard.
Wrexham had the first real chance of the second half after Mark Carrington had a 25-yard effort caught by Davies.
On the 58-minute mark, Vose curled a 30-yard effort high and wide on Davies’s post but shortly after, Wrexham did make it 3-1 through Sean Newton whose effort hit the post before he got up quickly to fire in the rebound from close range.
The visitors added a fourth through Smith who headed home a corner from the right at the back post.
Andy Mangan hit a 30-yard free kick which was saved by the legs of Taylor before Norwood had a low effort palmed away from eight yards.
On 82 minutes, Norwood did get a goal back to give Rovers a bit of hope. He did well, cutting inside from the left and slamming the ball from 12 yards into Taylor’s bottom left corner.
But it was not enough as Wrexham progress into round two of the competition.
Tranmere: Davies, Vaughan, McNulty, Ihiekwe, Ridehalgh, Mekki, Maynard, Jennings, Kirby (Mangan 67), Tomlinson (Dawson 67 (Sutton 90+2)), Norwood. Substitutes: Rutter, Gumbs.
Wrexham: Taylor, Newton, Smith, Hudson, Fyfield, Vose, Gray (Cofie 70), Jennings, Moke (Carrington 34), York (O'Brien 80), White. Substitutes: Evans, Caton.
Referee: Tom Nield
Yellow cards: Newton (39), Vaughan (56), Maynard (90+1), Taylor (90+3)
Attendance: 3397 (813)