Arsenal battled to a 2-0 win at Monaco but the victory was not enough to keep them in the Champions League.
The odds had been stacked against the English side following their 3-1 home defeat in the first leg, a deficit from which no team in the Champions League era has recovered when heading into the away second leg.
Olivier Giroud's first-half effort and Aaron Ramsey's goal after 79 minutes set up a grandstand finale, but the valiant Gunners just fell short as they bowed out on the away-goals rule.
After a slow start, Giroud headed a half-chance wide after 13 minutes. Arsenal were lacking a spark as Monaco – who had not conceded a single goal in their three home Champions League group stages earlier this season – happily took the tempo out of the match.
Giroud, who had missed several chances in the first leg, grabbed a lifeline for the visitors with a well-taken finish nine minutes before half-time.
Monaco continued to ride their luck as Arsenal pressed, with Mesut Ozil's free-kick tipped over at the start of the second half.
The game then opened up as Arsenal committed men forwards, and finally reduced the arrears further through substitute Ramsey after 79 minutes.
There was, though, to be no miracle of Monte Carlo as Monaco goalkeeper Danijel Subasic denied a late header from Sanchez to end Arsenal's European dreams at the last-16 stage once again – and leave Arsene Wenger's men counting the cost of conceding a stoppage-time third goal at the Emirates Stadium.
England forward Danny Welbeck started in support of Sanchez and main striker Giroud, with Theo Walcott again on the bench as manager Wenger named an attacking side on his return to the club he led to the French league championship in 1988.
Despite Arsenal's positive intentions, they remained vulnerable on the counter-attack, with Joao Moutinho curling an early 20-yard effort over following a quick break by the home side.
Officially only around 1,500 Arsenal fans had made the journey to the French Riviera, but given tickets were put on general sale, there were more than double that around the Stade Louis II.
Arsenal, though, struggled to find any sort of early flow as Monaco slowed the tempo, which suited the game of veteran former Tottenham and Manchester United striker Dimitar Berbatov.
As the quarter-hour mark approached, Arsenal full-back Hector Bellerin got away down the right and whipped the ball into the six-yard box, from which Giroud glanced just wide.
Arsenal centre-half Laurent Koscielny cannoned the ball up against the crossbar from close range after a deep free-kick into the Monaco penalty area, but the offside flag was raised.
Arsenal continued to look for an opening, but were frustrated by some solid defending from the hosts as the game. The only edge, at that stage, was the U2 rocker of that name watching on from the stands, alongside Bono.
The breakthrough finally came after 36 minutes.
After left-back Nacho Monreal had done well to keep the ball in on the far side, Danny Welbeck darted into the penalty area.
The loose ball fell for Giroud, and his shot was saved but cannoned up against his face, with the French striker alert to sweep in the rebound.
Arsenal almost grabbed a quickfire second when Welbeck's fierce, low shot hit Aymen Abdennour on the inside of his leg as the defender was sprawled on the ground.
Sanchez was shown a yellow card by Norwegian referee Svein Oddvar Moen for tumbling in the penalty area as Nabil Dirar touched him on the back.
In first-half stoppage-time, Welbeck's low cross from the right flicked up off Giroud's knee and into the grateful arms of goalkeeper Subasic.
The Croatian was in the right place again at the start of the second half to acrobatically tip over a curling 20-yard free-kick from Ozil.
Although Monaco did not need to score, they still carried a threat on the counter as Arsenal pressed a high line.
Wenger made a positive change after 63 minutes as Ramsey replaced holding midfielder Francis Coquelin.
Ozil flashed a low drive wide as Arsenal again created a good opening which lacked a telling finish.
With 20 minutes left and two goals still needed, Welbeck was replaced by Walcott.
Arsenal's pressure finally told when Ramsey made it 2-0 with 11 minutes left.
Monreal darted into the left side of the Monaco penalty area, before cutting the ball back to Walcott, whose low shot came back off the base of the post.
Layvin Kurzawa should have cleared but stabbed the ball to Ramsey, who cracked it into the bottom left corner to set up a grandstand finale.
Subasic clawed Sanchez's close-range header off the line, but despite pouring forward during five minutes of stoppage-time, Arsenal were again left to reflect on what might have been.
The Monaco bench celebrated wildly as the principality club reached a first Champions League quarter-final since 2003/04.
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