Sanchez strike sends 10-man Arsenal through
Arsenal's Chilean striker Alexis Sanchez (R) shoots on goal during a match in London on August 27, 2014 - by Ben Stansall
The Chilean, Arsenal's number 17, repaid the first instalment of his £30 million ($50.4 million, 37.6 million euros) transfer fee by tucking away a low shot in first-half stoppage time, but the hosts' inability to kill the game off made for a night of nail-biting tension at the Emirates Stadium.
Mathieu Debuchy's dismissal for two yellow cards meant that Arsenal saw out the last 15 minutes with 10 men, but although a solitary goal would have sent the Turkish side through after last week's goalless draw in Istanbul, the hosts held on for a 1-0 aggregate win.
"We produced the performance we wanted -- on the tactical side, on the technical side, and on the mental side," said Arsenal manager Arsene Wenger, who revealed that striker Olivier Giroud will be out for up to four months after undergoing surgery on a broken tibia.
"We were at the level that was requested tonight against a good team, but we couldn't finish the game off and that made it nervous in the last 10 minutes when we were down to 10 men."
Wenger's side, who lost to Bayern Munich in the last 16 last season, will discover the identities of their group-phase opponents in Thursday's draw.
"Of course we are gutted, because we played good in both games and are close to qualifying for the Champions League against great opponents," Besiktas coach Slaven Bilic told Sky Sports.
"Those couple of chances we had to score, but we didn't. Congratulations to Arsenal. They had the experience and quality up front when needed."
Arsenal were missing Aaron Ramsey through suspension, while Giroud's absence saw Sanchez handed another opportunity to prove he can lead the line on his own.
Though Arsenal initially took control, their chances owed more to serendipity than intricate build-up play.
- Sanchez makes breakthrough -
Jack Wilshere slid a shot wide after Sanchez had been bundled over, while Santi Cazorla should have done better than slam the ball wide from 40 yards when visiting goalkeeper Tolga Zengin fluffed a kick.
Besiktas, meanwhile, had two penalty appeals waved away, first when Mustafa Pektemek tumbled under a challenge from Debuchy and then when Wilshere needlessly slid in from behind on Ramon Motta.
Reprieved, Wilshere then set Sanchez free, only for Zengin to save at the former Barcelona forward's feet, but in first-half injury time the same two players combined to put Arsenal ahead.
When Debuchy's cross was headed clear, Wilshere toed the ball to Mesut Ozil and although the England midfielder could not take the German's return pass into his stride, the ball broke kindly for Sanchez to roll a low shot past Zengin.
Eager to make their place in the group-phase safe, Arsenal started the second period strongly and forced the visitors onto the back foot.
Cazorla drilled the ball across goal from an Alex Oxlade-Chamberlain pass, Sanchez poked wide after being neatly teed up by Wilshere, and Debuchy headed wide at the back post from an in-swinging Cazorla free-kick.
The visitors remained a threat, however, and after Debuchy had alertly thwarted Demba Ba, the former Chelsea striker lashed a shot into the side-netting.
Oxlade-Chamberlain should have doubled Arsenal's lead when he shot straight at Zengin from Nacho Monreal's low cross and the hosts' task was further complicated when Debuchy saw red after tugging back Pektemek.
In a moment of farce, Wenger introduced defender Calum Chambers in place of Ozil, only for the former Southampton youngster to be booked by Portuguese official Pedro Proenca for entering the field of play too quickly.
As Bilic watched animatedly from the stands following his dismissal in the first leg, Besiktas pushed hard in the closing stages, but with Ba failing to apply a touch to a late cross, they slipped into the Europa League.