Having built a 4-1 lead in the first leg in north London last week it seemed Arsenal had plenty of room for error in the CSKA Arena but they were wobbling when Fedor Chalov and Kirill Nababkin scored to give the hosts hope of a memorable fightback.
Salzburg beat Lazio 6-5 on aggregate, scoring four goals in 20 minutes of the second half to edge out the Italians.
Arsenal would have to improve to get through the semi-finals, where they will meet one of Marseille, Atletico Madrid or Salzburg.
It finished 6-3 on aggregate with Wenger's side now preparing to face either Atletico Madrid, RB Salzburg or Marseille in the next round. "I think we missed a bit of movement to be available for our team-mates, we didn't win the duels, so it was hard".
"We tried to improve the game level and improve our performance thanks to the substitutions but after 75 minutes, the tempo of the match began going down and down".
The visitors scored a second when Jean-Kevin Augustin's superb free kick turned the tie in Leipzig's favour again in the 55th minute.
GETTYCSKA Moscow are notorious for violence and unrest
Leipzig hit back through Jean-Kevin Augustin, and at that point they would have progressed on away goals, only for Dimitri Payet and Hiroki Sakai to score and make sure of Marseille's progress via a 5-3 aggregate success.
Marseille, who are fourth in Ligue 1, have reached their first European semi-final in 14 years.
"You could see in Europe a lot of turnovers".
Sporting made Diego Simeone's Atletico side suffer in the Portguese capital where Fredy Montero's 28th minute goal halved the Spanish club's first-leg lead.
The Premier League side must now await their fate in Friday's semi-final draw in Nyon, as they aim to keep their challenge for the trophy, and an automatic spot in next season's Champions League group stage, alive.
Trump may allow states to drug test food stamp recipients
The testing would apply mostly to able-bodied adults and those who are applying for specialized jobs, according to an AP report. In one state, 321 tested positive at a cost of almost $2,650 per positive test.