The Frenchman went down under pressure from Sebastien Bassong and then picked himself up to give the Gunners a 2-1 lead early in extra-time after the north London derby finished 1-1 after 90 minutes.
Nasri then scored a second penalty - after Steven Caulker pulled down Marouane Chamakh - and Andrey Arshavin completed the scoring in a 4-1 victory for Arsene Wenger's men.
Redknapp said: "I thought the first penalty was harsh. He fell over. Really, I think he dived for the first penalty.
"The second was blatant."
Bassong agreed on the first spot-kick, adding: "He went down too easily, I didn't think it was a penalty.
"He told me afterwards that I barely touched him.
"It leaves a bad taste in my mouth. I touched Nasri's chest but I did not grab him. I have seen the video and from my point of view, it was not a penalty."
Nasri revealed he had agreed in advance with manager Arsene Wenger that he would take any spot-kicks at White Hart Lane, after Tomas Rosicky missed from 12 yards against Sunderland on Saturday.
"I provoked the foul and then I didn't take the penalty," Nasri said. "But after I talked with the boss and said, 'Next time I will take the penalty', and I scored."
Wenger explained that Nasri had previously resisted taking penalties after fouls on himself.
"He had a superstition to think that when a penalty is made on you not to take it," Wenger said. "But I tried to prove to him that that's solely based on fear.
"He realises now that it was a bad superstition."
Nasri hailed Jack Wilshere as a player with a big England future after
the Arsenal youngster's outstanding midfield display inspired his side to victory.
"He's a really good player," Nasri told Sky Sports.
"Last year when he went (on loan) to Bolton it helped him because he played a lot of games.
"Now he's ready to play and be a good player for Arsenal.
"And I think England need a player like him. Jack is a wonderful prospect for the future."
Wilshere is hoping Arsenal can go all the way to Wembley in the competition, but cautioned: "We don't want to get ahead of ourselves.
"In the last couple of years, we fell short.
"Hopefully we can concentrate on each match as they come and hopefully we can win it."