Schalke had to recover from a 2-0 first leg defeat to the Ajax youngsters and did it to force extra time with Leon Goretzka and Guido Burgstaller.
LYON and Besiktas have been handed two year suspended bans by Uefa after their European clash was marred by crowd trouble.
In the final minute of normal time Lacazette hit the crossbar for the visitors following a fine through-ball from Corentin Tolisso.
After 10 ideal penalties the two sides could still not be separated, but Lyon keeper Lopes was the first to make a save when he denied Tosic but Jallet was unable to capitalise as he sent his spot kick high over the bar. But Lopes saved again from Matej Mitrovic, allowing Gonalons to calmly convert and send Lyon through.
United, the last remaining English side in Europe, took the lead in the 10th minute through Henrikh Mkhitaryan but Sofiane Hanni brought Anderlecht, winners of both the old European Cup Winners' Cup and the UEFA Cup, back level in the 32nd minute.
Talisca's brace (27', 58') kept the Turkish giants in the tie, with Alexandre Lacazette having earlier restored parity. They took a 3-0 lead on the night, only for two late extra-time goals to swing the tie in Ajax's favour.
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But Besiktas chairman Fikret Orman said that Lyon had nothing to be scared of at the team's stadium on the shores of the Bosphorus, distancing the club's fans from responsibility in the violence.
Leandro Trossard levelled just four minutes later but it was not enough to dent the Spaniards' charge to the last four.
Yet Ajax never abandoned their slick passing game and grabbed an away goal when Viergever controlled a cross from Kenny Tete and managed to fire into the roof of the net with nine minutes left.
Lacazette went agonisingly close to doubling his tally four minutes prior to the interval, turning away in despair as his curling effort struck the inside of the right-hand post before rebounding across the face of goal and behind.
Lyon nearly produced another swift riposte as Lacazette thumped an effort straight at Fabri when played through and the Besiktas keeper - at fault for Lyon's victor last week - produced a superb save after Lucas Tousart's shot deflected heavily off Atiba Hutchinson.
Either side of the interval, Diakhaby headed wide and Lopes clawed out Tosun's effort before Fabri scrambled across his line to brilliantly deny Tolisso - the last action of note before Lopes and Gonalons broke Besiktas hearts.