The Red Devils scored their first four kicks through Ryan Giggs, Carlos Tevez, Cristiano Ronaldo and Anderson to capitalise after Jamie O’Hara was denied by Ben Foster and David Bentley fired wide.
United could have won the game inside the 90 minutes but Ronaldo’s injury-time drive came back off the inside of the post, while the Portugal international was harshly booked for diving in the 67th minute when he should have won a penalty following a Ledley King challenge.
Patrice Evra shot over just before the end of extra-time while Spurs’ best chances fell to Aaron Lennon and Darren Bent.
On 71 minutes Foster was forced to beat away a vicious drive from Lennon before denying bent with his legs from 10 yards with 126 minutes on the clock.
But Sir Alex Ferguson’s men kept their nerve in the shoot-out – as they did in the Champions League final against Chelsea last May – to lift the first domestic honour of the campaign and stay in the hunt for a clean sweep.
Colchester will hand late fitness tests to Ryan Jackson and Brandon Comley prior to their clash with Yeovil.
Arsenal secured their place in the quarter-finals of the Europa League following their 3-1 defeat of AC Milan at the Emirates Stadium on Thursday.
Danny Welbeck scored twice against AC Milan to toast his England recall by sending Arsenal into the last eight of the Europa League.
Ipswich striker Joe Garner will not play again this season after it emerged he suffered a fractured skull against Sheffield United last week.