Bolton goalkeeper Adam Bogdan was in inspired form and denied dominant Spurs repeatedly until Ryan Nelsen finally broke the deadlock with a 74th-minute header.
Gareth Bale added the second three minutes later. Louis Saha came off the bench to wrap up the victory for Spurs after Kevin Davies had scored a late consolation for Bolton.
"We couldn't have played any better than we did. It was so one-sided," Redknapp said.
"Their keeper was amazing. I have never seen a keeper make so many saves. He made save after save, we hit the woodwork. We had an incredible amount of chances.
"I always felt the goal would come.
"I am looking forward to a great (semi-finals) weekend. They will be two great games, a London derby and a Merseyside derby. You wouldn't like to pick a winner. It is wide open."
The match was preceded by a minute of applause for Fabrice Muamba, who has made "encouraging progress" since he collapsed but remains in intensive care.
"We are not quite there yet but it has been a miracle to see the boy recover as he has," Redknapp said.
"Everybody's spirits couldn't be better, when he left here that night we all feared the worst. We keep getting good reports every day and that is just great news.
"It was an horrendous night. Those of us that were here will never forget it. Please God he makes a recovery like we hope he will."