United overturned a 2-0 first leg deficit for just the third time in their history thanks to a Robin van Persie hat-trick at Old Trafford.
Manager David Moyes went into the match on the back of a 3-0 defeat to Liverpool, which raised questions about his future at the club.
The Scot has endured a woeful debut season domestically, but Evra thinks some pride has been restored with the dramatic comeback win.
"The Champions League is a tough competition and everyone thought we were out (before kick-off), but we showed Manchester United are still alive," said Evra, who skippered the side in the absence of groin injury victim Nemanja Vidic.
United are no strangers to comebacks, even though they had only overturned a two-goal deficit twice before Wednesday's game.
The Red Devils came back from 2-0 down to beat Barcelona 30 years ago, they turned the tables on Roma with a 7-1 thrashing in 2007 and no United fan will ever forget those two decisive goals by Ole Gunnar Solskjaer and Teddy Sheringham in the 1999 final.
Having joined the club in 2006, Evra played in just one of those three games but he is aware of the club's history of defying the odds.
"Maybe people will say I am too old school, but when you have the Manchester United spirit you can make the difference and nothing is impossible," the Frenchman said.
"I am really pleased for everyone, for the fans, for the new staff, for the players - it was really important to show the love and respect for the shirt and we won because of the spirit."
Evra felt United owed the fans another memorable European night after Sunday's humbling against old foes Liverpool.
"When I got home after the Liverpool game I was so sad," the 32-year-old said.
"I was happy the fans supported us until the end, but I was so hurt inside and couldn't wait for this game to give something back. We've had a tough season and they are fantastic."