Spurs dominated the first-half but City keeper Shay Given was in magnificent form to deny Jermaine Defoe, Roman Pavlyuchenko and Ledley King.
Defoe finally broke the deadlock on 29 minutes but City came out a different side after the break and substitute Valeri Bojinov equalised on 65 minutes.
But Spurs then won it with four minutes to go after substitute Frazier Campbell tangled with Micah Richards in the box and Robbie Keane made no mistake from the penalty spot.
Redknapp told Sky Sports that Spurs should have been well in front before City came back into the game.
Great opportunities"We should have come in well clear - we had some great opportunities, played some great football first-half - could have come in two or three up for sure," said Redknapp, adding he then lost Jonathan Woodgate at half-time and had to re-organise, before Jermaine Jenas was ill.
"It never got started the second-half and we suddenly lost our rhythm, they came on stronger and then it was game on then. They got a goal back and really we're struggling to get a grip of the game then but we got the penalty in the end - that sewed it up," said Redknapp, who thought the penalty was deserved from where he was sitting.
"It looked a nailed on penalty to me - it looked like he climbed all over him," said the Spurs boss, adding the ball did not need to be in playing distance for the penalty to be awarded.
Redknapp was pleased with the new club record of just ten goals conceded at home all season.
"The lads have done well from where we came fromto pick up the amount of points we have has been a fantastic achievement by the players," he said.
Fulham favouritesDespite the win, Redknapp believes Fulham - who are two points above Spurs - are in the driving seat to finish seventh and qualify for Europe via the league.
"We have got to go to Liverpool for a very tough game so you would have to see Fulham as red-hot favourites," he said, adding his side will give it their best shot against the Reds.
"But wherever we finish, seventh or eighth now, will be a terrific achievement," said Redknapp, who received a standing ovation from the fans after the final home game of the season.
He praised both the players and the fans for their efforts since he came to the club.
"They've done brilliant - we've had a great season, we've turned it around fantastically well. It's been enjoyable and the fans have been terrific and so have the players - they've responded really well to me and I've been delighted with them."