Arsene Wenger's side were paired with the Turks at the draw in Switzerland, while Shakhter Karagandy of Kazakhstan stand between Celtic and a place in the group stage.
Arsenal's draw is complicated by Fenerbahce's ongoing battle to avoid expulsion from this season's competition.
The Istanbul club were banned from European football for two seasons in June over the involvement of club officials in match-fixing but won the right to continue their qualifying campaign pending a Court of Arbitration for Sport appeal.
A CAS decision is expected before the draw for the group stage is made on August 29, but in the meantime Fenerbahce will host Arsenal in the first leg of their play-off on Wednesday, August 21 before the return leg in London the following Tuesday.
No decision has yet been made on what would happen if Fenerbahce were to beat Arsenal and subsequently lose their appeal.
Celtic, meanwhile, were in the half of the draw reserved for national champions and must travel to Kazakhstan for the first leg of their play-off on Tuesday 20 August.
Shakhter have come through two qualifying rounds to reach the final stage and count former group stage participants BATE Borisov of Belarus among their beaten opponents.
The return match at Celtic Park will be played on 28 August.
Elsewhere, the standout tie pairs PSV with AC Milan in a repeat of the 2005 semi-final.
Play-off draw - champions route
Dinamo Zagreb v Austria Vienna
Ludogorets Razgrad v Basel
Viktoria Plzen v Maribor
Shakhter Karagandy v Celtic
Steaua Bucharest v Legia Warsaw
Play-off draw - non-champions route
Lyon v Real Sociedad
Schalke v Metalist Kharkiv
Pacos de Ferreira v Zenit St Petersburg
PSV Eindhoven v Milan
Fenerbahce v Arsenal