The Gunners will fly to Istanbul on Monday following a dramatic stoppage-time 2-1 win over Crystal Palace on the first day of the new Barclays Premier League season.
There is, however, little opportunity for Wenger's squad to regroup before their focus turns to extending the record of reaching Europe's elite club competition for each of the last 16 campaigns.
Qualification would bring with it some 8.6 million euros (£7million) in prize money, with each subsequent group home match worth around £3m of additional matchday revenue.
Wenger knows all too well what is on the line against Slaven Bilic's side, who defeated Feyenoord 5-2 on aggregate in the third round of qualifying when former Chelsea striker Demba Ba scored a hat-trick in the second leg.
"We are desperate to do it because we want to play in the Champions League and we know how big these games are," said Wenger.
"We are desperate to go through. No matter what it costs we want to go through.
"We always have a difficult draw, we had Fenerbahce last year, this time we have Besiktas, so it is difficult."
Wenger added: "The preparation for us was a Premier League game we had absolutely to win, that's the best preparation for confidence.
"Now the preparation will be mainly to recover and to study Besiktas."
With Besiktas' Vodafone Arena ground being redeveloped, Tuesday night's match will take place at the 72,000-seater Ataturk Olympic Stadium, where Liverpool famously beat AC Milan in the 2005 Champions League final.
Reports in Turkey suggest the home fans are preparing a hot reception for Wenger, whom they feel was critical of the club following their 8-0 defeat at Liverpool during the 2007/08 group stages.
"They are specialists at doing that. I never said that. They always create things like that. I am used to playing Turkish teams and I never said that," Wenger replied.
Captain Mikel Arteta believes the players will relish the intimidating atmosphere.
"They will be in our face, but I like playing in atmospheres like that," he said.
"It is different to what we are used to here in England and it's attractive to the players.
"The pressure is on us to get through because they are expecting us to get through but we are facing a tough opponent with good players and experience in the Champions League, so it won't be easy there."
Wenger confirmed: "It was our plan to bring the Germans back in contention for Everton, but in case of emergency I will take Mertesacker.
"It is a bit early for him. Ideally we would need one more week preparation, but he is willing to play, no problem."
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