Alexis Sanchez could eventually cost Arsenal as much as £35 million. Against Besiktas, he repaid a chunk of that fee by scoring the goal that took Arsenal through this tie and into the Champions League proper.
Qualification for the group stage is said to be worth around £25 million. For Arsenal's morale, it's priceless.
The thought of relegation to the Europa League is unpalatable to most Arsenal fans. They regard that as Tottenham territory. They needn't worry: their Thursday nights are safe. Arsenal have reached the group stage for the 17th consecutive year.
Much of the prematch chatter had surrounded the unavailability of Olivier Giroud. The France international was injured in rather comical fashion against Everton, blocking the ball with his foot. However, there's nothing funny about the consequences: Arsene Wenger confirmed today that he will be out until late December at the earliest with a broken foot.
In Giroud's absence, Arsene Wenger again selected summer signing Alexis Sanchez as his central striker. Sanchez had started in that role against Everton, before being withdrawn after an ineffective first half. However, Wenger clearly believes the Chilean can thrive through the middle, and was undeterred by the apparent failure of his experiment. Despite the return to fitness of Yaya Sanogo, the Arsenal boss plumped for Sanchez as his "No. 9." By halftime, it looked like a shrewd move.
Without Aaron Ramsey and Giroud, the big question was just who would score the goals Arsenal needed to take them into the Champions League group stage. Jack Wilshere was the first to go close, screwing an effort wide after bursting through on goal. Santi Cazorla was next, lobbing a stranded goalkeeper but missing the posts in doing so.
Shortly before the half-time whistle, the breakthrough arrived, and it was Alexis who provided it. Jack Wilshere kept a cleared ball alive and played a one-two with Mesut Ozil before bursting into the box. As the ball returned his way, Wilshere took a touch before astutely ducking out to leave the chance to the Chilean. He finished calmly across the goalkeeper to open his Arsenal account.
Alexis had endured a difficult half, struggling to make his presence felt at the point of the Arsenal attack. However, his movement and clinical finishing demonstrated why Wenger seems determined to persist playing him as a central striker. He's not yet an accomplished centre-forward, but he clearly possesses all the requisite attributes to be one eventually.
Arsenal's lead was a little harsh on Besiktas, who had acquitted themselves well for the first 45 minutes. The Turkish team induced two particularly heart-stopping moments in the Arsenal penalty box. In the first incident, Mathieu Debuchy was required to produce a quite remarkable last-ditch challenge. Later, Jack Wilshere's attempt to do the same ought to have resulted in the award of a penalty. Somehow, the Englishman and Arsenal escaped punishment.
The second half was a very tense affair, particularly when Debuchy was dismissed for a harsh second booking. The Frenchman had been treading a tightrope after being booked in the first half, and was unfortunate to be given his marching orders for what looked like a clean tackle.
As the pressure built, Alexis emerged as an outlet for Arsenal. His work rate was extraordinary, as he showcased the same formidable spirit that saw him lead the line so effectively for Chile in the World Cup.
In the second half, it was plain to see why Wenger believes he could one day be a top centre-forward. He has strength, skill and speed. Crucially, he also has courage. Against Besiktas, Sanchez showed the heart that will inevitably be required to cope with the regular buffeting he will receive from Premier League defenders.
Thanks to his goal tonight, Sanchez will also have to cope with Champions League defenders. On the evidence of this all-action display, that is a challenge he will meet head on.