The outcome will no doubt be even more frustrating for Rovers, who were leading 1-0 on Saturday in the original tie when the game was abandoned due to floodlight failure!
The scoring was opened as early as the 7th minute on Monday as Moors` Jamey Osborne received the ball in midfield and strode forward to beat Tranmere `keeper Luke Pilling from 35 yards.
The on 25 minutes, Pilling was a little fortunate not to have been embarrassed when he went over to near the corner flag to clear the ball, only for it to go straight to home defender Jamie Reckord, whose `Beckhamesque` effort went narrowly wide.
But 5 minutes before half-time, the Moors did double their advantage when Fiacre Kelleher was adjudged to have been pushed in the box and Jermaine Hylton cheekily chipped the resulting penalty straight down the middle of Pilling`s goal.
Just shy of the hour-mark, Solihull almost made it 3-0 when a poor back-pass from Steve McNulty was pounced upon by Hylton, but his shot was brilliantly tipped round the post by Pilling.
Little was seen of the Tranmere attack, with a Connor Jennings header and an Ollie Norburn shot being the closest they came to pulling a goal back.
The game did end on a sour note for the home side, however, when they had Hylton sent-off for a late tackle on Adam Buxton, but it came too late to have a major effect on the game.
So, it will be Solihull who will travel to meet the winners of the Spennymoor Town and Gainsborough Trinity tie in the Second Round on January 13th.