The Gers took the lead in the 10th minute when Waghorn's shot from outside the box nestled into the bottom right corner.
Tavernier doubled Rangers' lead just three minutes later with a trademark free kick into the top right corner.
It was not all Rangers though, and Alloa had a rare venture into the attacking half after 30 minutes. The ball found its way to Michael Doyle but the midfielder's shot was blocked.
Veteran striker Michael Chopra then gave Rangers another scare in the 40th minute when his powerful shot bounced off the crossbar.
Waghorn made it three moments later when his shot hit the roof of the net after Barrie McKay played him in on goal.body check tags ::