The Austrians had broken the back of the tie with a 3-0 win in Amsterdam last week and completed the job 3-1 on home turf.
All the goals arrived after half-time with a Mike van der Hoorn own goal followed by strikes from Sadio Mane and Jonathan Soriano.
Salzburg would have opened the scoring earlier but for Jasper Cillessen in the Ajax goal.
His performance was about all the Dutch side had to savour, although Davy Klaassen did manage the most consolation of goals on 82 minutes.
Salzburg will meet Swiss champions Basle in the next round.
Ajax coach Frank de Boer kept faith with the majority of the starting XI that crashed to last week's heavy defeat, but again they made little impact.
Salzburg made all the running in a goalless first half. Cillessen made a string of good saves, his first coming after just six minutes to deny Alan's 20-yard strike.
Kevin Kampl then shot against the crossbar before Cillessen made a superb stop to keep out Soriano, who had scored a double in Amsterdam.
Twenty minutes had still not passed when Cillessen again came to Ajax's rescue, tipping away an Alan shot from the edge of the area.
Ajax eventually began to settle, reaching the break on level terms, but the tie was done and dusted soon after the break when Van der Hoorn put through his own goal.
Cillessen was this time left with no chance as the defender deflected Soriano's driven cross from the right beyond his goalkeeper.
Salzburg added another two goals inside the next 20 minutes - through Mane and Soriano - to add further gloss to a memorable scoreline.
Mane slipped the ball under Cillessen while Soriano increased his competition-best goalscoring record to seven.
Klaasen grabbed a late goal when his 18-yard shot deflected off a defender and the crossbar before going in.