Arsenal battled to a 2-0 win over Bayern Munich in the Allianz Arena, but still bowed out of the Champions League.
Trailing 3-1 from the first leg, manager Arsene Wenger accepted it would be a Mission Impossible for his beleaguered side in Bavaria and so it proved despite a heroic performance to beat the runaway Bundesliga leaders.
Oliver Giroud fired in after just three minutes and Bayern were kept at bay by some fine saves from stand-in goalkeeper Lukasz Fabianski, preferred to fellow Pole Wojciech Szczesny.
Defender Laurent Koscielny headed in to level the tie on aggregate with four minutes left, but there was to be no fairytale ending as the Gunners were knocked out on the away goals rule.
Bayern went into the tie on the back of formidable defensive displays and with an unbeaten home European record against English teams since Norwich won at the Olympicstadion in the 1993 Uefa Cup.
Thomas Vermaelen, the Arsenal captain, had been dropped to the bench, while England left-back Kieran Gibbs returned from his thigh injury.
The English Premier League side produced the perfect start as they swept into the lead on three minutes.
A slip from defender David Alaba allowed Tomas Rosicky to feed Theo Walcott on the right, and his low cross through the six-yard box was smashed into the roof of the net by Giroud.
Bayern are not used to trailing, having conceded just 10 goals in total in the Bundesliga so far, and were soon back on the offensive as Toni Kroos tested Lukasz Fabianski with a 25-yard drive.
Luiz Gustavo, in for the suspended Bastian Schweinsteiger then fired over after the ball was only half-cleared from a corner.
Arsenal, though, weathered the storm and maintained their own threat on the counter, with Walcott the main outlet down the right and Rosicky seeing more of the ball in a central role.
When Arjen Robben ran at the Arsenal defence, he was held off by Carl Jenkinson, standing in for injured right-back Bacary Sagna.
Walcott whipped over another decent cross through the German's penalty area, but this time Giroud was unable to connect as he raced in at the back post.
There was a stoppage on 41 minutes when Giroud was caught by Daniel Van Buyten, jarring his ankle and knee. Despite coming back on after treatment, the France forward was hobbling badly.
Gibbs went into notebook of Czech referee Pavel Kralovec when he slid in on Robben, despite appearing to take the ball cleanly.
Arsenal needed to score another two goals to stand any chance of pulling off the unlikeliest of comebacks, but it was Bayern who opened up the second half strongly as Robben and Kroos both fired just wide from the edge of the box.
Thomas Muller drilled a low shot under Fabianski and through the six-yard box as the Germans pressed again.
There was brief respite when Rosicky was fouled some 25 yards out, but Santi Cazorla hit his free-kick straight into the wall.
Bayern continued to press as Gustavo curled the ball just wide from the edge of a busy Arsenal penalty area.
Fabianski then stood up well to block after Robben had darted clear of the Arsenal defence.
With 20 minutes left, Wenger made a double change as Walcott and Aaron Ramsey were replaced by Alex Oxlade-Chamberlain and Gervinho.
The Ivory Coast forward, so often maligned for his inconsistencies, almost produced a moment of magic when playing a one-two with Cazorla in the right side of the penalty area, turning his marker to stab the ball across the face of goal and just wide of the far post.
Fabianski produced another good stop from Muller as Bayern again went in search of an equaliser.
However, it was Arsenal who set up a grandstand finish as Koscielny headed in from a corner, with a melee following in the goalmouth a goalkeeper Manuel Neuer laid on the ball.
Despite plenty of pressure, another clear cut chance never came and it was the Germans who progressed through to the quarter-finals as Wenger was again left to reflect on what might have been.
Arsenal will end an eighth consecutive season without silverware after their, leaving manager Arsene Wenger to rue a home first-leg defeat after his side almost pulled off an improbable comeback.
"I believe we were very, very close," Wenger told Sky Sports. "The spirit and the performance was there and we had a great go. The overall defensive performance was very good. They defended very very well and we lost some balls offensively in situations.
"It was frustrating sometimes but I put that down to the fact that they defended well and when we got the second goal, you could see that if we had put some more balls in the box we would have score another goal."
No team in the Champions League has recovered to qualify after losing the first leg at home by a two-goal margin and Arsenal were knocked out in the last 16 for the third season in a row.
Wenger pointed to Bayern's late third goal at The Emirates last month, scored in the 77th minute by Mario Mandzukic, as the hammer blow. "The regrets we have tonight of course is from the first leg when you think that we conceded a cheap goal. Had we lost 2-1 at home you see how important that goal was tonight," he said.
Asked about his team's performance in the absence of Jack Wilshere, Lukas Podolski, Sagna and Szczesny, Wenger said he took pride from their display in defeating last season's beaten finalists.
"I know these players. I have said many times they have a fantastic attitude and spirit and we have the quality. But unfortunately we have to go out of the Champions League and it hurts very much."
Arsenal almost pulled off a huge comeback in the Champions League round of 16 last season, losing to AC Milan 4-0 in the first leg and narrowly going out after winning the return 3-0.
"We have a bit of the same regrets we had last year in the last 16 - first game nerves and afterwards, we have no choice and show how good we are," added Wenger. "This is a 180-minute game."
Arsenal now face a battle to secure Champions League football next season by finishing in the Premier League top four. With 10 games left, they sit in fifth place, five points behind Chelsea in fourth.
"I'm confident that we will give absolutely everything to be back next season and keep the basis of this team," Wenger said. "You could see that the young players that played tonight are doing very well, let's finish well this season."
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