Olivier Giroud was the hero for Arsenal, scoring a hat trick to lead the London club to a 3-0 win and into the knockout phase at Olympiacos' expense.
Arsenal went into the sixth and final Group F matchday three points adrift of second-placed Olympiacos and knowing it had to win by two clear goals to leapfrog the host.
And it all went according to plan in Greece, as Giroud ensured Arsenal joined Bayern Munich in the round of 16 with two second-half goals after opening the scoring in the 29th minute.
Speaking to BT Sport post match, Wenger was beaming.
"I told you before the game that it would be the greatest escape," he said. "I'm very proud of the team because we needed to be right mentally and tactically and at 2-0 we needed to push for a third because it was dangerous.
"We had the perfect game. Nine times out of 10 you are out [after such a poor start] and we needed something special and we needed a complete team performance, and it was.
"This is a team, and they showed it again."
The manager also praised the hat-trick hero, saying: "He [Giroud] has shown character. The penalty was an important penalty and I'm happy for him, he did well again."
Arsenal's remarkable display avoided the club being eliminated from the group stage of the Champions League for the first time since 1999-2000.
"I'm proud of the consistency because it demands a lot of effort every day and that's what it's about," Wenger added. "We have escaped and maybe it's a lucky year for us in the Champions League.
"If you look at the players who are out and the performance we produced, it was outstanding. I would like to pay tribute to those who were not here and those who stepped in."