The Austrians fought back despite having Stefan Schwab sent off for a shocking two-footed lunge at Jairo Riedewald in the 59th minute of the third qualifying round first-leg tie.
Two other former European champions were in action on Wednesday with Celtic beating Qarabag 1-0 at home and Steaua Bucharest drawing 1-1 with Partizan Belgrade, also at home.
Four-times European champions Ajax, who beat AC Milan at the Ernst Happel stadium in the 1995 final, dominated the first half and two goals from Davy Klaassen put them in a commanding position at halftime.
Klaassen turned in Daley Sinkgraven's pass in the 25th minute and headed another two minutes before the break.
Florian Kainz got ahead of his marker to pull one back early in the second half and Robert Beric equalised in the 76th minute after a mix-up in the Ajax defence.
Dedryck Boyata struck in the 82nd minute for Celtic, turning in a Kris Commons cross, after they had toiled to break down the well-organised Azeri champions in Glasgow.
Fernando Varela snatched an 81st-minute equaliser for Steaua at home to Partizan after Miroslav Vulicevic had given the Serbs the lead just after the hour.
Swiss champions FC Basel, who reached the last 16 last season, won 3-1 at Lech Poznan where both teams finished with 10 men.
Michael Lang gave the visitors the lead in the 34th minute only for Denis Thomalla to level one minute later.
Shkelzen Gashi, topscorer in the Swiss league last season, missed a penalty for Basel midway through the second half after Tomasz Kedziora had been sent off for the hosts.
Marc Janko and Davide Calla scored in the last 15 minutes for the visitors, who then had Taulant Xhaka sent off.
Austrian champions Salzburg, who have failed to get through the qualifying rounds on any of their last seven attempts, overcame Malmo 2-0 at home and HJK Helsinki were held 0-0 by Kazakh champions FC Astana.
(Editing by Toby Davis)
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