Manchester United made the last eight with a 1-0 win over Rostov on Thursday at Old Trafford, going through 2-1 on aggregate.
Besiktas progressed to the quarter-final stage by beating Greek side Olympiacos 5-2 on aggregate.
As Spain's last representative in the competition, Celta Vigo will now face Genk, while Ajax will do battle over two legs with Schalke for a place in the last four.
Schalke, trailing at halftime to a deflected shot from Andreas Christensen and a 30-metre rocket from Mahmoud Dahoud, were helped by a bumpy pitch as they hit back against Gladbach.
Mourinho's team is now scheduled to play West Bromwich Albion (April 1), Everton (April 4), Sunderland (April 9), Anderlecht (April 13), Chelsea (April 16), Anderlecht again (April 20), Burnley (April 23) and Swansea City (April 30).
However, midfielder Paul Pogba was forced off with an injury in the second half, putting a dampener on the win.
Mouctar Diakhaby gave Lyon the lead in the 16th minute before Kevin Strootman equalised for Roma a minute later.
Their only previous quarter-final tie in this competition came in the 1984/85 UEFA Cup when Ron Atkinson's side lost on penalties to Videoton.
Racing Genk coasted through 6-3 on aggregate against fellow Belgians Gent after a 1-1 draw.