Talisca scored the opening goal of a pulsating match after 27 minutes before Alexandre Lacazette ensured Lyon, 2-1 up from the first leg, were back in charge of the tie with his fine finish moments later.
But Talisca added his second of the night shortly before the hour mark to take the game to extra time, and latterly penalties, only for Lyon to prevail after captain Maxime Gonalons converted the decisive spot-kick.
Joining Lyon in the draw for the last four are Ajax after they secured a dramatic 4-3 aggregate win against Schalke despite being reduced to 10 men.
Schalke looked to have pulled off a sensational comeback after being 2-0 down following the first leg in Amsterdam. They took a 3-0 lead on the night, only for two late extra-time goals to swing the tie in Ajax's favour.
Nick Viergever and Amin Younes scored the crucial goals for Ajax in the 111th and 120th minutes, after Daniel Caligiuri's headed goal in the first half of extra-time appeared to have won it for Schalke.
In Belgium, Pione Sisto's solo goal sealed Celta Vigo's place alongside United and Ajax after they edged out Genk.
Holding only a narrow 3-2 lead from the home leg, Celta kept their opponents at bay with relative ease for more than an hour before Sisto fired them in front on the night with a thunderous strike from the edge of the area.
Leandro Trossard quickly pulled one back to leave the Belgians needing only one more to take the overall lead on away goals, but the visitors stood firm to make the last four.