David Bentley also dragged his effort wide for Spurs before Anderson fired the winning penalty to make it a perfect weekend for United after Liverpool's defeat by Middlesbrough on Saturday left them seven points clear in the Premier League.
After adding the League Cup to FIFA's Club World Cup in Japan, United are still chasing a clean sweep of five trophies but manager Alex Ferguson played down the quintuple chances.
"We will keep our feet on the ground," Ferguson told reporters. "That's just a media thing. We could go to Fulham next week and a deflected shot goes in off someone's backside and we will be out of the FA Cup."
Ferguson made special mention of Foster, one of several reserves in the United team. He was the busier of the two keepers, saving well from Aaron Lennon in normal time, Darren Bent in extra time and then O'Hara's penalty.
"Without Ben Foster we would have lost today," Ferguson said. "He will be England's keeper for many seasons if he has a bit more luck with injuries."
HIGH-PROFILE ABSENTEES
United, with Dimitar Berbatov, Wayne Rooney and Michael Carrick not even in the squad, were the better side early on.
Cristiano Ronaldo fired a free kick over from 25 metres and Paul Scholes launched a superb long pass for Danny Welbeck but the teenaged reserve striker was closed down.
Ferguson beefed up United's midfield with Brazilian Anderson in the second half as the game meandered towards extra time.
Ronaldo was booked for diving, although replays suggested Spurs skipper Ledley King was lucky not to concede a penalty.
Tottenham improved as the half wore on and Lennon had their best chance with 20 minutes to go.
The ball was worked to him on the edge of the area and after setting himself and taking aim, he thumped a shot that Foster did well to beat away past the despairing lunge of Bent.
Ronaldo almost won it in stoppage time, his shot cannoning off the post and Carlos Tevez hitting the rebound wide.
Tottenham won the trophy last season against Chelsea in extra time and had the chance to repeat that feat when Bent found space but Foster saved his shot with his legs.
As penalties loomed Modric jinked through United's defence before a last-ditch tackle and Patrice Evra crashed a half-volley over the crossbar.
United had happy memories of their last penalty shoot-out, having clinched the Champions League against Chelsea on spot kicks, and there was an inevitability about the conclusion once O'Hara's effort was turned away by Foster.
Liverpool's latest slip-up means United are well placed to complete a hat-trick of league titles. Tottenham, on the other hand, have just a relegation fight to look forward to.
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