City won 2-1 on Friday to clinch the tie 4-2 on aggregate and book a berth in the competition's makeshift knockout tournament in Lisbon.
Pep Guardiola's side will face Lyon in a one-off tie on Aug. 15.
All Champions League games from the quarterfinal stage onward will be played in Lisbon due to the coronavirus pandemic. The tournament will follow a single-elimination format, up to and including the final on Aug. 23.
It's the first Champions League elimination for Zinedine Zidane as a manager. The Frenchman led Los Blancos to the title in 2016, 2017, and 2018.
City put themselves in a strong position to advance after winning the first leg 2-1 in February. They looked relatively comfortable in the reverse fixture, with the two sides playing 163 days after they first locked horns.
Jesus robbed Varane in his area and fed Sterling for the opening goal in the ninth minute. Karim Benzema equalized off Rodrygo's pinpoint cross to score his 65th Champions League goal, moving the striker into fourth in the all-time scoring list, but Jesus later pounced on Varane's misplaced header to restore City's lead.
City are now three games away from winning an elusive Champions League title. But Lyon are no walkovers. Les Gones took four points off City in last season's group stage, and prompted Guardiola to call them "one of the toughest teams I've ever faced."