Spurs managed to keep out the Italian giants on a nervy evening in north London to reach the last eight having taken a 1-0 lead from the first leg thanks to Peter Crouch's strike.
"This is an impossible dream that we have achieved so far," said Redknapp.
Tottenham have enjoyed a remarkable debut season in the competition having beaten reigning champions Inter Milan in the group stages before seeing off the seven-time European champions over two hard-fought legs.
The change is a remarkable turnaround from two and a half years ago when Redknapp took over with Spurs rooted to the bottom of the Barclays Premier League table.
He now faces the mouth-watering prospect of his side drawing a huge showdown with the likes of Real Madrid or Barcelona in the next stage next month.
Redknapp continued: "Two years ago if you would have said that we would have made it to the quarter-finals of the Champions League, I would have thought you were crazy.
"It's our first year in the club's history we have ever made Champions League football so that was a fantastic achievement to have won our group and to do what we have done against AC Milan.
"We have beaten AC Milan over two legs with two clean sheets and won our group, which was the hardest of the lot. It was always going to be nervy night. If you think you are going to smash AC Milan out of sight then obviously you haven't been watching football over the last few years.
"Sandro was immense in midfield, for a young lad with limited experience of the Champions League. We defended for our lives and the keeper did well for us too."
Source: PA
Source: PA