Jonas Eidevall's side were forced to settle for a 2-2 draw in the first leg at Borehamwood, but a second-half strike from Vivianne Miedema got the job done in a 1-0 second-leg success in Amsterdam.
With Beth Mead only considered for a spot on the bench after her North London derby exploits, the first half was largely devoid of attacking quality at both ends of the pitch.
Stina Blackstenius - who was drafted in for Mead - had a couple of half-chances that were comfortably gathered by Lize Kop, and Arsenal also lost centre-back Rafaelle Souza to injury towards the end of the first half.
Lotte Wubben-Moy replaced the Brazilian newcomer as Arsenal regrouped at half time, and it only took six minutes of the second period for Miedema to put the North London club in the ascendancy.
The Netherlands international - who scored a brace in the weekend's 4-0 win over Tottenham Hotspur at the Emirates - drove forward following a throw in and hammered a low effort into the far corner from the edge of the D.
Mead was soon introduced into the fray, but Ajax should have been level with 72 minutes on the clock, as Romee Leuchter's cheeky backheel freed Nadine Noordam down the left-hand side of the box, and her cross met the boot of Tiny Hoekstra who could only tap wide on the stretch from less than a yard out.
Frustrations were beginning to show for the home side, and Arsenal lost Mead to a nasty clash of heads with Lisa Doorn in the dying embers of the match before navigating six minutes of added time to progress.
Eidevall's side now face a two-week hiatus due to the international break and will be back in Women's Super League action away to Reading on October 16.