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."
Ferguson beefed up United's midfield with Brazilian Anderson in the second half as the game meandered towards extra time.
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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