Lampard got off to a losing start on his return to the hot seat at the club in a caretaker capacity on Saturday, with the Blues beaten 1-0 at Wolverhampton Wanderers.
Chelsea have had a disappointing season and are languishing in 11th place in the Premier League, having sacked manager Graham Potter on April 2 before bringing Lampard back.
"The Premier League is one of the greatest challenges in world football ... and those challenges have obviously taken us to a place we don't want to be as Chelsea," Lampard told a news conference on Tuesday, ahead of his second game in charge.
"I think the Champions League sometimes offers you a bit of escapism from that - a different speed of game, knockout football, those things can all contribute to getting different success in the same season.
"There's pressure on both teams due to the size of both clubs. Are Real Madrid favourites? Yes. (But) there's no better carrot in football than trying to prove people wrong.
"I think that's a nice challenge for us. I'm not worried about that, in terms of pressure. Football at this level is pressure and if you can't handle it, you're not a big club or big player."
Chelsea are 17 points off the top-four with eight Premier League games left so winning the Champions League is their only realistic road to Europe's elite competition for next season.
Last term they also faced Real Madrid in the quarter-finals and won the return leg at the Santiago Bernabeu 3-2 but were knocked out on aggregate due to a 3-1 loss at home.
"One thing I would say that the players have to understand is that this is a special arena and special football club," Lampard added. "My focus will not be on last year, it will be on what we can do this year as a team.
"I think we have to understand the strength of the Real Madrid team. We have to be disciplined with our game off the ball, understand the threats, ways they like to score and how they control the game.
"We also have to show our own strengths in our game. We have to have belief and follow through with what the idea is under pressure."
The Chelsea boss said midfielders N'Golo Kante (32) and Mason Mount (24) plus defender Thiago Silva (38), who all missed the trip to Wolves but trained with the squad on Tuesday, will be available.
"They're all fit, they're all in the squad. They are good and they are here," Lampard said.
