Everton goalkeeper Tim Howard saved kicks from Dimitar Berbatov and Rio Ferdinand before Phil Jagielka converted the one that counted for the Toffees.
Everton will play Chelsea, 2-1 winners against Arsenal on Saturday, in the final at Wembley on May 30.
United, who have already won the League Cup and Club World Cup already this season, now have to focus on retaining their Premier League and Champions League crowns.
In a game of few chances, United, who fielded an under-strength side, felt they should have had a penalty in the second half when Danny Welbeck was clipped by Jagielka, but referee Mike Riley infuriated Sir Alex Ferguson by ignoring their appeals.
Cahill blazed Everton´s first spot-kick high over the bar. But Howard responded by saving Berbatov´s weak effort and then pushing away Ferdinand´s shot before Jagielka slotted home the decisive kick to stun United.
"It was a tough game for us," Moyes said.
"If they put out their under-14s it would have a been a tough game - they are going after every trophy.
"For everyone at Everton it is fantastic. If they keep doing what they are doing now it, success will not be too far away."
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