Fedor Chalov and Kirill Nababkin struck in the opening 50 minutes to threaten a dramatic turnaround in the tie, but Welbeck and Ramsey scored in the final quarter of an hour to seal a 6-3 aggregate victory.
In a week that has seen Roma shock Barcelona and Juventus almost beat Real Madrid in remarkable Champions League quarter-final matches, much of the pre-match talk in Russia focused on the need for Arsenal to put in a professional performance to defend their 4-1 first-leg advantage.
They were punished for slack defending in each half through Chalov and Nababkin, though, as Viktor Goncharenko's side showed signs they would punish the Gunners' complacency and snatch a spot in the last four.
But Welbeck, starting wide on the left in place of the injured Henrikh Mkhitaryan, started and finished Arsenal's first particularly slick move before Ramsey netted on the counter-attack deep into injury time.
CSKA looked unlikely to threaten the latest European comeback in a tepid first half-hour, with the best of the early opportunities falling to Alexandre Lacazette, who was denied by a good block from Aleksei Berezutski.
The home side were dealt a blow when the creative Alan Dzagoev was forced off with injury 38 minutes in, but they were given some hope in the tie barely 90 seconds later.
Chalov reacted quickest to turn in the rebound after Petr Cech saved Nababkin firm downward header.
The Gunners goalkeeper would have hoped for a quicker response from his defenders after making the save, and there was a case of deja vu five minutes into the second half, when Nababkin beat Hector Bellerin to the loose ball to fire home after Cech had parried Aleksandr Golovin's drive from 30 yards.
The visitors responded with a spell of possession, but Mohamed Elneny's headed finish was correctly ruled out for offside at the end of one of their few promising attacking moves by that stage.
Wenger brought on Calum Chambers for the ineffective Jack Wilshere in a bid to strengthen his side's defensive set-up, but more lacklustre work in their own box allowed 38-year-old Sergei Ignashevich to volley inches wide of the left-hand post.
However, just as CSKA threatened again, they were stunned by a brilliant piece of play from Welbeck. The England forward spun into space down the left, found Elneny and latched on to the Egyptian's superb throughball before curling a shot past Igor Akinfeev.
10 - Danny Welbeck has reached double figures for goals in all competitions for the third time in his career (also 2011-12 and 2013-14). Relief.— OptaJoe (@OptaJoe) April 12, 2018
The goal quietened the atmosphere at the CSKA Arena and Arsenal saw out the closing minutes with ease.
Akinfeev denied Welbeck a second with a fine save, and Ramsey dispatched a good finish with practically the last kick of the game after Elneny played him into clear space on the break.
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.
Key Opta stats:
- Arsenal have reached their first major European semi-final since 2008-09, when they did so in the Champions League.
- The Gunners are the fifth English team to reach the semi-finals of the Europa League, after Fulham (2009-10), Liverpool (2009-10 and 2015-16), Chelsea (2012-13) and Man Utd (2016-17).
- CSKA are now winless in eight European matches against English opposition (D2 L6).
- Aaron Ramsey has had a hand in eight goals in his last eight matches for Arsenal in all competitions (seven goals, one assist).
- Arsenal have conceded 31 goals in all competitions in 2018, three more than any other Premier League side.
- Danny Welbeck’s goal took him to double figures for goals in all competitions this season, the third time in his career he has reached this landmark after 2011-12 and 2013-14.