Spurs won 1-0 at Eastlands through Peter Crouch's header to secure fourth spot and at least a place in the qualifying rounds of Europe's elite club competition.
Tottenham have not played in the European Cup since their legendary double-winning side lost in the semi-finals to Benfica in 1962.
Redknapp, who was doused with an ice bucket by his celebrating players, said: It was great to come here and win - I took a big gamble tonight and played an attacking team. We went for it.
"People will think I'm mad coming away from home like this - all the top teams only play with one up front now.
"I'm very pleased - it's great for the club, a good achievement and we have played great football this year."
The success marks a remarkable 18 months for Redknapp who took over in October 2008 when Spurs were bottom of the Premier League with two points from eight games.
Redknapp added: "Daniel Levy [Spurs chairman] was scared that we were going to get relegated, that's why he made the move.
"At the time he was afraid we could have been stuck in a relegation battle and we were in it for a long while.
"Of course this means a lot to me, I'm delighted but I know I'm good at my job, I wouldn't have done it if I was a mug."
Redknapp picked out Crouch, fellow striker Jermain Defoe and defenders Ledley King and Michael Dawson for special praise.
"King and Dawson - my God they were good, Crouchie led the line unbelievably, it was a great performance from him, and the best we have seen from Defoe for a while," said the Spurs boss, who does not hold out much hope of Arsenal losing against Fulham on Sunday to allow Tottenham to sneak into third.
"I just wanted to make sure we got there and finished fourth, the chairman has just said to me 'who have Arsenal got on Sunday' but I doubt very much whether that will happen.
"I'm just very happy to finish fourth, that's where we wanted to be.
"It was important someone got in from outside the big four I could only see Man City breaking in.
"I thought it was going to be really tough - I could only see us finishing in the top seven and getting a European place.
"That says an awful lot for us."
Manchester City manager Roberto Mancini tried to end speculation over his future at the club by insisting he will be at Eastlands next season.
Asked whether he was confident about remaining manager, Mancini said: "Yes I'm confident. I don't know but I think I stay here. Why not?
"I work here four or five months and I think when you build a house you don't start from the roof but the basement. We work very well but we are near the roof.
"I am not a magician, I don't have a magic wand. We wanted this (fourth) place and we tried like Liverpool, Tottenham and Aston Villa.
"When you don't finish fourth this is football but we did a good job."
Mancini claimed that City had been unlucky to lose tonight, and in key games against Manchester United and Everton.
He added: "It's disappointing that we lost this game - we didn't deserve to, we have been very unlucky in some games.
"At the same time we must be proud because we had a good season, even if we are not playing in the Champions League next year.
"Now we have a good team and can improve on next season."
Mancini said failure to make the Champions League would have no impact on the future of Carlos Tevez, who has had issues with the Italian coach over his training regime.
"Carlos Tevez has a contract with Manchester City and he will be here next season," said Mancini.