United wasted a hatful of opportunites before Sulley Muntari's matchwinning penalty and Redknapp said: "When I saw United's team on the notice board, I was a bit surprised because I thought they would rest Ronaldo and (Wayne) Rooney.
"I told my lads they were good players but they needed to go out and prove it. To their credit, they worked their socks off.
"We knew it would be a tough game as it was a draw no-one wanted.
"We came here a few weeks ago, lost 2-0 and it could have been 10.
"Of course we rode our luck today. You know United are going to get chances as they are a fantastic team.
"You have to take your opportunities when they present themselves."
Kuszczak was sent off and replaced by skipper Rio Ferdinand. However, the defender was given no chance with Muntari's cool spot-kick.
It is the third time Redknapp has enjoyed beating United in the FA Cup.
He did it first of all for minnows Bournemouth in 1984 and against with West Ham in 2001.
This is the first time the veteran boss has advanced beyond the quarter-finals but said: "I am not getting too carried away.
"I have had a number of beatings from United over the years, don't forget.
"I was hoping Sulley would just hit the target as I knew Rio at West Ham and he is not the best goalkeeper.
"It is marvellous for our supporters and the club we are going to Wembley for the semis. It will be their greatest day for years.
"It probably means more to our fans than those of Manchester United as you can get a bit blasé about these things."