Hodgson had called for belief ahead of his side's last 16 second leg tie at Craven Cottage on Thursday night despite trailing 3-1 to the Italian giants.
His words looked to have fallen on deaf ears when the Cottagers went behind in the second minute to a David Trezeguet strike.
However, they produced a stunning fightback with goals from Zoltan Gera (2) and Bobby Zamora before a stunning late match-winning chip from substitute Clint Dempsey secured a 4-1 win and a 5-4 aggregate success.
"I'm on top of the world," beamed Hodgson. "I don't know if it is the biggest night in the club's history but it must come close.
"I was very impressed with how they came back as we couldn't have had a worse start.
"We went behind to a simple goal which made an uphill task even steeper but even from that point the team started to play well and the crowd got behind us and even at half-time I thought we were looking good to win the game.
"It was never going to be impossible to turn it around and luckily that has happened."