In their debut Champions League campaign, Spurs beat the seven-times winners 1-0 on aggregate after a tense goalless draw at White Hart Lane and have their sights set on the final at Wembley.
Tottenham 0 AC Milan 0 (agg 1-0): Read Matt Lawton's match report
Gallas glory: the Spurs defender (second right) is mobbed by team-mates after a superb display
Redknapp plans to celebrate with a bacon butty and a cuppa and hopes to avoid Arsenal's conquerors Barcelona in next week's quarter-final draw.
"We've given the fans something to be proud of," said the Tottenham manager. "What we've achieved is an impossible dream. Nobody could've seen this coming. We've made the last eight and if you'd said that two years ago, they'd have thought you were crazy.
"I'll enjoy it but I won't get carried away. I don't know how far we can go. I've had ups and downs and I'm just looking forward to getting home and having a bacon sandwich, a cup of tea and taking my dogs out. It's all about the fans. They're already dreaming. It's the first time we've ever played in the Champions League.
"It's a fantastic achievement by the players and we deserve to be here. We won our group, tough as it was, and we beat AC Milan and didn't concede a goal to them."
Last line of defence: Gallas clears fromunder the bar
After winning the first leg in the San Siro, three weeks ago, Spurs had to defend for their lives last night against the Serie A leaders and their manager reserved special praise for central defenders William Gallas and Michael Dawson and young midfielder Sandro, who he said was "immense".
Redknapp added: 'It was always going to be a nervy night. If people thought we were going to come here and smash AC Milan out of sight, they've not been watching football over the years. It was difficult to get close to them at times.
'Sometimes a lead like that puts you in a difficult situation. We played with width and we got out-numbered because they squeezed it in midfield and played with a diamond. I thought Clarence Seedorf was amazing.
'We're playing AC Milan here, we're not playing Raggy A*** Rovers. We got a clean sheet and that was important. We're normally a team who scores and lets goals in.'
Rock on: Defenders Gallas and Dawson celebrate at the final whistle
Schalke, Barcelona and Shakhtar Donetsk are through with Spurs with the final four to be decided next week but everyone will hope to avoid favourites Barca in the draw.
'They were amazing against Arsenal,' said Redknapp. 'They made a great team look so ordinary and didn't give them a shot at goal. It was one of the finest performances I've seen. They were on another level. Arsenal looked out of their depth.
'It's going to be very difficult for any team to beat Barcelona this year but we want to stay in this competition and go as far as we can.'
Milan manager Massimiliano Allegri said: 'It leaves a bitter taste. We deserved more. But the performance was encouraging. We are top of the league and will do utmost to stay there.'
Mathieu Flamini said: 'We spent the entire match in Tottenham's half, looking for the goal. Everyone saw we played a great game with the right spirit. We came here to earn qualification and we did not, so we are sad. We gave our all and the goal did not come.'
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Explore more:People: Michael Dawson, William Gallas, Harry Redknapp Places: Barcelona, Milan, Europe