After going in 2-0 down at halftime, Medhi Carcela pulled one back for Liege before Renaud Emond scored a 94th minute penalty to equalise after Ajax striker Klaas-Jan Huntelaar and Dusan Tadic put the four-time Dutch European Cup winners ahead.
Visitors Ajax had looked comfortable after former Southampton winger Tadic set up the first goal with a swerving cross before drilling home the second himself.
However Belgian side Liege clawed their way back into the game and Emond struck the woodwork before slotting home from the spot after Noussair Mazraoui fouled substitute Moussa Djenepo.
The winners of the tie in Amsterdam next week will face either Slavia Prague or Dynamo Kiev in the playoff round for a place in the group stage of Europe's elite club competition.
Those teams drew 1-1 in the Czech capital after Josef Jusbauer's 95th minute penalty for a dominant Slavia cancelled out Benjamin Verbic's 82nd minute opener.
In Lisbon, Benfica earned a narrow victory in a match they controlled as Franco Cervi's deflected strike after 69 minutes handed them the advantage over Fenerbahce, with both teams entering in this round of the qualifying process.
When the teams played in the European Cup first round in 1975 Benfica won 7-0 but this time the Turkish side can hope to stage a second-leg comeback to secure a playoff against Thessaloniki-based PAOK or Spartak Moscow who meet on Wednesday.
Stanislav Dragun's header earned Belarusians BATE Borisov a 1-0 win at last season's Champions League group qualifiers Qarabag, with the eventual winners facing PSV Eindhoven.
Dinamo Zagreb won 2-0 at Astana, who reached the group stage in 2015, with goals from Mario Budimir and the impressive Dani Olmo to put one foot in the playoffs where Young Boys await.
Elsewhere, Malmo drew 1-1 with Vidi and Red Star Belgrade were held by Spartak Trnava with the same scoreline.
(This story has been corrected to make clear PAOK is in Thessaloniki not Athens in para 8.)
(Reporting by Rik Sharma; Editing by Ken Ferris)
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