For the second year running the Premier League side have been drawn against Turkish opposition in the play-off round, with Wenger hoping to repeat last season's 5-0 drubbing of Fenerbahce.
Arsenal completely dominated that tie a year ago and they return to Istanbul on Tuesday aiming to do the same on the back of a dramatic stoppage-time 2-1 win over Crystal Palace on Saturday.
Besiktas will provide stern opposition, though, having already flexed their muscles in the competition with a 5-2 aggregate victory over Feyenoord in the third qualifying round.
Wenger has concerns about the recovery time afforded to his squad ahead of Tuesday's clash following their Premier League exploits, but he remains convinced that the importance of the tie will ensure his players are fully focused.
"We are desperate to do it because we want to play in the Champions League and we know how big these games are," Wenger told the club's official website. "We had to give a lot [against Crystal Palace], hopefully we will recover and be ready for Tuesday night.
"No matter what it costs we want to go through. We have always a difficult draw, we had Fenerbahce last year, this time we have Besiktas. It's difficult."
Whether Laurent Koscielny will make the trip remains uncertain after Wenger revealed a foot problem had flared up following the win over Palace.
If he is not fit, Per Mertesacker could be rushed back into action, having been given an extended break along with countrymen Mesut Ozil and Lukas Podolski after their World Cup-winning exploits in Brazil.
"We will assess Koscielny on Monday," Wenger added. "If he's fit and capable to play, we play him.
"It was our plan to bring the Germans back in contention for Everton. But in case of emergency I will take Mertesacker."
