Vidic's claim for wrongful dismissal was rejected by an independent regulatory commission on Tuesday.
He was sent off in United's 3-1 defeat on Sunday for a foul on Eden Hazard.
An FA statement read: "The player's three-match suspension commences with immediate effect."
The 32-year-old Red Devils captain will now miss the League Cup semi-final second leg against Sunderland on Wednesday and Barclays Premier League games at home to Cardiff and away to Stoke.
His absence is a blow to David Moyes' struggling side, whose season suffered another setback in their comprehensive defeat by Chelsea, which left them 14 points off league leaders Arsenal and six off the Champions League places.
The Serbian will be missed by a defence which has looked particularly vulnerable in recent matches, having conceded nine goals in five games.
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- Barclays Premier League
- David Moyes