Besiktas piled on the pressure to try and level the tie on aggregate and it finally told 10 minutes into the second half when Talisca's powerful header from Adriano's cross made it 2-1 on the night and 3-3 overall.
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.
Mitrovic then saw his penalty saved by Lopes - and when Gonalons scored it meant Lyon would be in the hat for Friday's semifinal draw.
The French and Turkish teams were excluded from taking part in the next continental club competition they qualify for but the governing body decided that the suspension will be deferred "for a probationary period of two years".
Lyon president Jean-Michel Aulas described the sanction as "fair".
There has been more supporter trouble for Lyon to experience since that night, with their Ligue 1 match at Bastia abandoned on Sunday when home fans entered the pitch to confront visiting players.
The tension was on the pitch rather than the stands as both sides scrapped energetically for the semi-final berth.
United Nations strongly condemns North Korean missile launch after delay
It also states that North Korea's "illegal ballistic missile activities are contributing to its development of nuclear weapons". Secretary-General Antonio Guterres will brief members on the North's nuclear program.
The return leg in Turkey takes place on Thursday (20:05 BST).
Roared on by the packed Vodafone Arena on the shores of the Bosphorus just opposite a former palace for Ottoman Sultans, Besiktas drew first blood as Brazilian Talisca scored after a corner in the 27th minute. Lacazette's exquisite touch - following fine interplay with Gonalons - allowed him to execute a ideal side-foot volley over Besiktas goalkeeper Fabricio.
Besiktas had the running for most of the extra time but failed to convert chances.
Anderson Talisca bagged a brace around Alexandre Lacazette's sensational lob, but Lacazette hit the woodwork twice, Ricardo Quaresma thumped the post in the final minutes and Ryan Babel missed a sitter. Fabricio denied Lyon's Corentin Tolisso with a spectacular one handed save.
A statement released by Uefa read: "The CEDB has made a decision to exclude Olympique Lyonnais [and Besiktas] from participating in the next Uefa club competition for which it will qualify".
The November 1993 clash between Istanbul's Galatasaray and Manchester United - when Turkish fans welcomed the English team with "Welcome to Hell" banners - is still remembered but Turkish football bosses like to think fan behaviour has moved on since then.