Spurs trail 4-0 from the first leg after Jose Mourinho's side took advantage of Peter Crouch's early dismissal to run riot at the Bernabeu.
But captain Dawson has already been involved in one European comeback this season and is desperate to ensure Tottenham's maiden Champions League campaign doesn't end on a whimper.
Red mist: Peter Crouch is sent off after just 15 minutes of Tottenham's first leg defeat to Real Madrid
'We'll never give up dreaming. I always say that,' he told the Evening Standard. 'When you're in the competition, you've got a chance. We know it is going to be hard. There's no hiding that. But we've certainly got players who can cause them some problems. They'll probably be saying the same about themselves but we'll be attacking. So it's going to be an open game.'
Tottenham's European adventure looked to be over before it began when they trailed 3-0 to Young Boys of Berne back in August.
But having pulled two goals back in Switzerland, Spurs completed the job at White Hart Lane with a thumping 4-0 win.
'Three down after 20 minutes, it certainly felt like it was all up,' Dawson admits. 'But we showed a lot of grit and determination to come back. Then, we got Inter Milan in the group, went 4-0 down and came back with the goals by Gareth [Bale].
'And then what a night we had against them at White Hart Lane [winning 3-1]. And we did it against AC Milan. All that shows that we can do it, eh? Let's hope we can do it again.'
Despite Crouch's red card in Spain and Aaron Lennon's withdrawal from the squad during the warm-up due to illness, Dawson denies that he and his teammates were overcome by the magnitude of the occasion.
Comeback kings: Tottenham came from 3-0 down to beat Young Boys Bernes to qualify for the Champions League group stages
'I wouldn't say the players were overawed, far from it. Going a goal down so early on didn't help and obviously going down to 10 men with Crouchy sent off didn't help either.
'Lenny had been ill since the Sunday. He just wasn't well enough to play and he was gutted to miss out on a night like that. We've all been ill in the past and had days off work before.'
Should Spurs fail to overcome the odds on Wednesday night, their only route back to the Champions League will be by finishing in the top four of the Premier League.
Just as they did last season, Harry Redknapp's side look set to battle it out with big-spending Manchester City for the final qualifying spot.
And Dawson is aware of the importance of playing in Europe's premier competition, particularly if the club wants to hold on to its star players.
'Our aim hasn't changed since the start of the season,' Dawson adds. 'We've got some big important games coming up and, hopefully, we'll get the right results and we'll be back in. We'll keep aiming for that fourth spot and we'll keep believing we can do it.
In his element: Harry Redknapp has led Tottenham from the bottom of the Premier League to the last eight of the Champions League
'I certainly hope the team stay together. Gareth's just signed a new contract, which is a big bonus. Rafa [van der Vaart] has been great this season. And Luka [Modric] is a fantastic player. He's not the biggest but he's strong and he just glides past players.'
Dawson was also full of praise for Redknapp, who has taken Spurs from the bottom of the Premier League to the upper echelons of the league.
And the suggestion that his manager is next in line to succeed Fabio Capello doesn't go down well with the defender.
'The gaffer leave for England? Oh, no, what he's achieved here is great,' Dawson says. 'He came in - it was eight games and two points or something like that. You look at the squad and there are a lot of players still here, myself included.
'We were lacking confidence and not playing to the best of our abilities. The gaffer made us believe in ourselves. First game we went out to Bolton and won and it just seemed to turn things round. Confidence and belief is a big thing.'
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?Explore more:People: Michael Dawson, Peter Crouch, Fabio Capello, Harry Redknapp, Aaron Lennon, Jose Mourinho Places: Madrid, Spain, United Kingdom, Switzerland, Europe
Source: Daily Mail
Source: Daily Mail