Assombalonga set the hosts on their way with the opener in the 38th minute of an all-League One affair at London Road when firing past Owain Fon Williams at his near post.
And he took his tally for the season to 14 with a couple of second-half crackers to clinch Posh's place in Sunday's draw - repeating the 5-0 league success they enjoyed over Rovers at Prenton Park in August.
Assombalonga cut into the box and guided a low shot past Fon Williams in the 64th minute before completing his treble nine minutes later with the pick of the bunch - a curling 20-yard drive after collecting a Lee Tomlin pass on the left wing and darting inside.
And it was also a great afternoon for strike partner Shaun Jeffers, who celebrated a first Posh start with his first two goals for the club.
He converted a Tomlin corner in first-half stoppage time and then swept home a Michael Bostwick cross in the dying moments.
Tranmere bowed out with a whimper as they failed to muster a single shot on target.