However, the hosts hit back nearly immediately, through Moanes Dabour to offer hope to the home fans.
With 19 minutes remaining Lazio still led 5-3 on aggregate but Salzburg scored three times in five insane minutes through Amadou Haidara, Hee Chan-hwang and Stefan Lainer to claim a remarkable 6-5 aggregate victory.
A shell-shocked Lazio couldn't respond, leaving Salzburg to progress 6-5 on aggregate on yet another truly memorable night of European football.
RB Leipzig travel to Marseille with a 1-0 advantage, while Salzburg host Lazio trailing 4-2 from the first leg.
It was another night of comebacks in a insane midweek in Europe, with Marseille and Red Bull Salzburg producing stirring performances to overturn first-leg deficits.
Fedor Chalov slotted home to give CSKA the lead five minutes before half time, and when Kirill Nababkin made it two five minutes after the restart, it looked as though the comeback might just be on.
However, Salzburg equalised a minute later through Munas Dabbur.
The Red Bulls produced another Roma-styled comeback, handing a 4-1 hiding to Lazio- Roma's neighbours- to qualify 6-5 on aggregate after losing the first leg 4-2.
But they did that before the interval in front of a huge Velodrome crowd, via a Stefan Ilsanker own goal, Bouna Sarr and Florian Thauvin.
Atletico Madrid are also through, and remain the favourites to win the trophy in Lyon next month - they were beaten 1-0 on the night away to Sporting Lisbon but progressed 2-1 on aggregate.
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.
Marseille, who are fourth in Ligue 1, have reached their first European semi-final in 14 years.
CSKA's fans, hundreds of whom watched shirtless from behind the goal, roared on their team but Danny Welbeck's well-taken goal calmed Arsenal's nerves.
Elneny then sent Ramsey - scorer of a brace in the first leg - away to score Arsenal's equaliser on the night in stoppage time.
Leipzig poured forward looking for the decisive away goal but Marseille's Hiroki Sakai broke clear in stoppage time to wrap up the victory, firing into an empty net after Leipzig keeper Peter Gulacsi had gone forward for a corner.
Meanwhile, Atletico held on in the rain in Lisbon to win 2-1 on aggregate against Sporting, despite losing on the night to a Fredy Montero header in the first half.
Atletico's possible opponents in Friday's semi-final draw are Salzburg, Arsenal and Marseille.