The 31-year-old admits that United damaged their reputation after last season's group stage exit from the competition but believes the side can regain their fear factor if they defeat Los Blancos in the last 16 tie.
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"Don't forget, last year we were knocked out in the first stage of the Champions League and maybe people have that image of Manchester United in Europe.
"We have to change that. That is why I think this is the perfect year to show who we really are. We can't miss this opportunity to show ourselves, it would be a massive disappointment.
"This is why I'm really looking forward to the match with Real Madrid. I've been waiting for a big European tie like this for a long time and it's probably the biggest European game since Barcelona in 2008.
"The fans have been waiting for this too. Manchester United v Real Madrid is a game for big players, big characters, big personalities."
United will first face a tough test against Everton before challenging Jose Mourinho's side, and then have a difficult FA Cup fifth-round tie against Reading, with Evra challenging his team-mates to step up to the plate as they chase a potential treble.ad>
"It's a really important time for us," he added. "It's important that we do well in the league in February, but we also have the Champions League game against Real Madrid.
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Sir Alex Ferguson recently remarked that the corresponding fixture last season against Everton, a 4-4 draw, cost his side the title, and Evra is also keen to atone for the Toffees' 1-0 over United at Goodison Park earlier this season.
"Nobody forgets the big disappointment of losing the title last season," he added.
"It's really important that we just keep going because the league is not finished in February. We have a lead but nothing is done yet.
"We just have to take it game by game and aim to win every match. That's the right attitude to have.
"In every game this season we have had the right attitude. That's why I think the fans and the players are very happy this year because we really want to show we are champions. We have to keep going that way."
Evra is also looking forward to the Red Devils summer tour of Asia. United will play one game in Yokohama, a city that holds special memories for the Frenchman after winning 2009's Club World Cup there.
“It was special [to win the trophy],” the Frenchman told the club's official website
“The fans were desperate to see United win that competition. And when we lifted that trophy, to be the world champion club, that was just amazing.
“I remember we played a friendly against Japan for France and the Japanese fans went crazy whenever Shinji touched the ball.
“It'll be the same as when Ji [Sung-Park] was in Korea – I always used to joke with him that there is God, then the King of Korea, and then Ji-sung Park! I think Shinji will be the same.”