Lampard's Derby County will take on Aston Villa at Wembley a week on Monday in the Championship play-off final, with a £180million bounty and top-flight football awaiting the winner on the other side.
Redknapp said: “I've seen everything from Frank that I expected — he's a very astute, clever guy.
“When I rang Mel Morris when the job became available, I told Mel what he would get if he took Frank and, I have to be honest, he took a bit of convincing at the time.
"But I arranged for him to meet Frank and, once he met him, he rang me straight away and said, 'You're spot on about everything you've said and I'm going to give him the job'.
“I'm pleased everything I told Mel about him has been what he got.
“Frank has been fantastic, he has been brilliant.
“He is bright as a button, knows the game, understands the game, he has been brought up with it."