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Arsene Wenger hailed Arsenal's "great escape" as Olivier Giroud's hat-trick at Olympiacos sent the Gunners into the Champions League knockout rounds.
Going into the game having lost three of their opening four Group F fixtures, Wenger's side needed to win by two goals or more at the Karaiskakis Stadium or better the 3-2 defeat they suffered at home to the Greek champions in September.
With seven key first-team players missing through injury, Giroud produced the goods when it mattered as Arsenal made light work of a seemingly difficult task to seal a 3-0 win.
Only seven sides in Champions League history had managed to progress from the group with three points or less after four games - but Arsenal have now added their name to an exclusive list with a result that left their manager beaming.
"I told you before the game it would be a great escape, and I must say I'm very proud of the team,'' he said. "We needed to be right on the mental side and right on the tactical side as well. We knew we had to push for the third goal because it's very dangerous at 2-0, and we had the perfect game.
"Nine times out of 10 in that position you are out of the group. We needed something special to switch it on and come back like we did tonight. We needed a complete team performance and that's what it was. Maybe it is a lucky year for us in the Champions League. You never know."
After a somewhat cagey start, Arsenal controlled much of the rest of the game at their pace, picking off Olympiacos at the right times.
Giroud opened the scoring when his near-post header squirmed past Roberto in the hosts' goal after great work from Mesut Ozil and Aaron Ramsey.
The France international then had a scare early in the second half as he went down with an ankle knock - but he soon dusted himself off and added the second.
This time it was great work from Joel Campbell, once a loanee at Olympiacos, as he controlled an Ozil pass before producing the perfect pass into Giroud for him to sweep home.
His hat-trick was complete when referee Nicola Rizzoli pointed to the penalty spot as Nacho Monreal's shot was blocked by the arm of the diving Omar Elabdellaoui.
Giroud duly despatched the resulting penalty and left Wenger purring at his display.
"It was a perfect performance," he said.
"He scored a header, one with a good combination thanks to a good pass by Campbell, and also a penalty on top of that.
"He was sometimes in uncomfortable positions but he fought like mad and he has shown skill and character.
"I always defended Olivier on the character side because he has always had something special on that front. He showed that in the game again today."
Arsenal could now face Andre Villas-Boas' Zenit St Petersburg, Manchester United's conquerors Wolfsburg or the Spanish trio of Real Madrid, Barcelona or Atletico Madrid in the next round.
Olympiacos, meanwhile, drop into the Europa League after failing to join Bayern Munich in qualifying from Group F.
Their head coach, Marco Silva, felt his side were left in a state of limbo after Giroud's second goal meant they needed to score themselves to advance.
"We started strongly, we tried to create chances," he said.
"In the second half I think we were influenced the second goal of Arsenal. The truth is that we fell somewhat below the strength of our team.
"It is true that at 2-0 we intended to take risks, to keep trying. We had the opportunities to score but our goal was at risk. The third goal killed our hopes and there ended the match. Whatever we did from that point we knew what it was too late."