Having overcome Lincoln Red Imps, Willem II and Galatasaray in the qualifying rounds – and fresh off an Old Firm triumph over Celtic – Steven Gerrard's side battled their way to three points in difficult conditions at Stade Maurice Dufrasne.
Tavernier scored the opening goal of the Group D contest in the 19th minute, Rangers' captain converting from the spot for his ninth goal in 10 outings this season.
The visitor relied on the crossbar to keep out a Jackson Muleka header before the break but were rarely troubled in a second half played out amid torrential rain, Roofe sealing a deserved victory when he spectacularly lobbed Arnaud Bodart from long range in stoppage time.
Connor Goldson had been the unlikely hero against Celtic, scoring twice in a 2-0 win at the home of their rivals, and the defender once again played a pivotal role with his aerial prowess.
It was his downward header from a corner that struck Nicolas Gavory on the arm, allowing the in-form Tavernier to tuck away the opportunity from 12 yards out.
Bodart was unable to keep the attempt out despite guessing the right way, though he did twice deny Alfredo Morelos early in the second half to keep the deficit down to one.
However, the goalkeeper was finally beaten again in the second added minute. Ex-Anderlecht forward Roofe pinched possession and, spotting the opportunity, lofted the ball spectacularly over the out-of-position Boudart from inside Rangers' half.