Kuszczak, who replaced Edwin van der Sar at half-time, was sent off and Rio Ferdinand took over in goal.
United went close to scoring, Sylvain Distin denying Michael Carrick on the line and Patrice Evra hitting a post.
The surprise win took Portsmouth back to Wembley for the first time since 1939 for the semi-finals and ended United's hopes of repeating their 1999 treble triumph. For Redknapp, who has previously knocked United out of the FA Cup with both Bournemouth and West Ham, it was another victory over Sir Alex Ferguson to savour.
Paul Scholes, Carlos Tevez and Owen Hargreaves all returned to United's starting line-up after the Champions League victory over Lyon - and they met with stubborn and well-organised Portsmouth resistance.
United were understandably furious after six minutes when Cristiano Ronaldo was clearly bundled over by Distin as he raced into the area, but referee Martin Atkinson ignored the appeals, to the clear disgust of Ferguson.