AC Milan advanced to the quarter-finals of the Champions League on Tuesday after Arsenal fell agonisingly short of a remarkable comeback despite an emphatic 3-0 victory over the Italian champions. Arsenal had been left needing to pull off one of the biggest comebacks in European history after suffering a 4-0 first leg thrashing against the Serie A leaders in the San Siro last month. But first-half goals from Laurent Koscielny, Tomas Rosicky and captain Robin van Persie fired the Gunners into an improbable 3-0 lead which left the Emirates faithful daring to dream that a miracle was on the cards. However Milan regrouped well after the break and Arsenal were unable to find the fourth goal that would have levelled the tie, leaving the shattered Premier League giants reflecting on a 4-3 aggregate defeat. Arsenal boss Arsene Wenger praised his side's display. "They can only be proud of their performance," Wenger said. "The team has fantastic spirit and showed some fight after the first tie, but unfortunately we go out. We had the chances for a goal, but we keep our winning run going which is important. We have paid the price for a bad first game. "But we have given everything. That is all you can do at the top level: give everything and then you accept the result." Milan coach Massimiliano Allegri meanwhile paid tribute to the way his side had responded after their first half mauling. "I just said to them at half-time that we couldn't change what had happened in the first half," Allegri said. "But I said to them it was important to remember that even at 3-0 we were still qualified. "I'm very happy because our objective was to qualify and we have done that." Arsenal had needed to make an early breakthrough to give their hopes of a miracle some momentum and they duly scored the goal they craved after only seven minutes to send the Emirates into raptures. Alex Oxlade-Chamberlain swung in an inviting corner from the left, and as the jittery Milan defence appeared to be preoccupied with the position of van Persie, Koscielny popped up unmarked to head home. The goal infused Arsenal with belief and within minutes they were sweeping forward hungrily once more. Theo Walcott surged down the right flank and fed van Persie, whose shot from a tight angle was well saved by Christian Abbiati. Arsenal's relentless pressing and constant harrying prevented Milan from settling into any kind of rhythm. Occasionally the home side fell foul of referee Damir Skomina, the Slovenian official showing yellow cards to Bacary Sagna and Kieran Gibbs for fouls on Mark van Bommel and Stephan El Shaarawy respectively. Yet Arsenal remained firmly in control and van Persie drew another good save from Abbiati with a vicious curling shot that was destined for the top corner. On 26 minutes the second goal that Arsenal had been threatening arrived when Walcott burst down the right and crossed low for van Persie. Thiago Silva intercepted but his weak clearance found Rosicky on the edge of the area and the Czech midfielder took his time and placed his shot low past Abbiati. Shellshocked, Milan desperately tried to steady the ship with half-time looming. But just when it looked like they had weathered the storm, Arsenal made it 3-0 two minutes before the break. Oxlade-Chamberlain popped up on the right flank and powered into the AC Milan penalty area. The youngster's pace took him through the gap presented by Antonio Nocerino and Djamel Mesbah and when the Milanese duo sandwiched the winger, Skomina looked long and hard before pointing to the spot. Van Persie stepped up to do the honours and put Arsenal within one goal of nirvana. But Milan returned from the break a far more organised unit, pressing Arsenal with a vigour that had been totally absent in the opening 45 minutes. Arsenal's best attempt to level the tie came on 59 minutes, when Gervinho raced clear down the left. The Ivorian's low shot took a deflection off Philippe Mexes and struck Abbiati's leg, but the ball bounced only as far as van Persie. For once though the Dutchman's killer touch deserted him, and instead of opting for power he attempted to lift the rebound over the Milan keeper who was able to save. At the other end, Arsenal lived dangerously, and were fortunate not to concede when Wojciech Szczesny's slack clearance fell to Zlatan Ibrahimovic, whose shot from distance flew just wide. Szczesny was to save Arsenal twice in quick succession in the final 15 minutes as the home side poured forward for a goal, blocking Ibrahimovic's low shot and then saving from Nocerino from point-blank range. Yet time ran out for the Gunners and Milan were through.