Held at Istanbul's Ataturk Olympic Stadium due to rebuilding work at the home side's own ground, the contest was hotly contested, with both sides coming close on several occasions.
Former Chelsea striker Demba Ba ran rampant throughout, and tested Gunners goalkeeper Wojciech Szczesny continuously, whilst Alex Oxlade-Chamberlain hit the post with Arsenal's best chance.
Wenger's plans for the return leg at the Emirates Stadium were hampered by a late red card for Arsenal midfielder Aaron Ramsey, forcing the Welshman out of the fixture.
The French manager will arrive in North London with a mixed bag in terms of lessons learned.
Here are five important points for Wenger to focus on for the second leg.
Calum Chambers is More Than a Deputy
The £16 million acquisition of Southampton's 19-year-old defender Calum Chambers this summer caused more than a few raised eyebrows by both the Arsenal faithful and those on the outside looking in.
A consistent, but unspectacular, performer for the Saints last season, Chambers was perhaps seen as one for the future.
However, in true Wenger fashion, the youngster has been thrust straight into action, and this contest at the Ataturk represented somewhat of a baptism of fire, serving as a Champions League debut for the versatile option.
With Chambers starting as partner to Laurent Koscielny at centre-back, in lieu of the rested Per Mertesacker, Jeremy Wilson of The Telegraph wrote how this fixture was the teenager's "biggest mental and physical test."
It is safe to say, however, that Chambers passed with flying colours, and Arsenal's clean sheet was much to do with the performance of the young defender.
With Mertesacker set to return to action in the near future, Chambers continues to pose Wenger an interesting selection dilemma.
Arsenal Need More Time to Adjust to Alexis Sanchez
The £35 million signing of Barcelona and Chile forward Alexis Sanchez this summer remains the most high-profile of the Premier League transfer window for the 2014/15 season.
Having scored 19 goals and made 10 assists in 34 La Liga appearances last season, Sanchez looks very much the missing piece in Wenger's meticulously detailed puzzle, and has impressed so far for the Gunners.
Against Besiktas, the 25-year-old was lively and creative, operating on the right flank for most of the contest.
Sanchez, as a strong, powerful, pacey attacking option, gives Arsenal another dimension in terms of a quick transition into the counter-attack, and this will no doubt prove devastating as the season progresses.
However, it was clear throughout the Gunners' draw with Besiktas that Sanchez's teammates are still adjusting to this frenetic style of play—Olivier Giroud in particular looked to struggle at times, not least in finding his shooting boots.
It will take time, but Wenger will surely find a way to mould his team around his strongest player.
Demba Ba Has a Point to Prove
Having spent four seasons in the Premier League, Besiktas forward Demba Ba left England on a rather sour note having been increasingly marginalised at Chelsea.
The Senegalese striker made just five starts last season under Jose Mourinho, with another 14 appearances coming from the bench; in return, Ba scored five goals.
Signing for the Turkish club in a £4.7 million deal this summer, Ba looked like a player with a point to prove in a first competitive encounter with Premier League opponents since the move.
This was not least evident by an audacious halfway-line attempt on goal that had Szczesny scrambling to tip the on-target striker onto the crossbar.
From the off, Ba was energetic and dangerous, coming closest to netting a winner for Besiktas with a rasping effort that fizzed just wide in the first half.
Returning to English soil next week for the return leg, Ba will surely continue to shine and will be one to keep an eye on.
Aaron Ramsey is Human After All
Last season saw the rise of Welsh midfielder Ramsey close to the heights that were projected on the player's move to North London from Cardiff City in 2008.
Scoring 12 goals and making a further eight assists in the Premier League and Champions League combined last season, Ramsey became central to every success the Gunners had.
In particular, the midfielder's partnership with Giroud blossomed to great effect—the Frenchman is at his very best alongside Ramsey.
Unfortunately, Wenger will have to make do without the angelic midfielder, who earned his first red card in senior football on Tuesday night.
Whilst the 23-year-old wasn't at his glittering best against Besiktas, the loss will be felt greatly in the return leg.
This is a sentiment held by Wenger, who described the decision as "a bit unbelievable," per BBC Sport.
Nevertheless, the Frenchman will have to adjust his plans to compensate for the loss of Ramsey in the Emirates fixture next week.
A Hard-Fought Draw is Not the End of the World
Although the Gunners would have like to secure an advantage in this fixture prior to the return leg, a hard-fought draw away to Besiktas is not to be snorted at.
Whilst the Ataturk stadium is not the Turkish side's home, the Besiktas faithful ensured an atmosphere akin to a Colosseum; a significant mental hindrance.
Coming through the tie without an away goal will be a disappointment, but a clean sheet should be considered second best.
Arsene Wenger and Arsenal can now look forward to next week's fixture on more comfortable home soil.
Should Arsenal have taken more from than a draw away to Besiktas? Let us know in the comments below.