Vidic was sent off in the first minute of stoppage time during United's 3-1 weekend defeat at Stamford Bridge following a tackle on Eden Hazard.
On Tuesday, the Football Association revealed that an Independent Regulatory Commission had decided to uphold referee Phil Dowd's decision to show Vidic a straight red card.
Vidic will now be unavailable for the second leg of United's League Cup semi-final with Sunderland on Wednesday, which Moyes' team approach with a 2-1 deficit from the first leg.
In addition, the 32-year-old will miss the Premier League visit of Cardiff City next Tuesday and a trip to Stoke City on February 1.
A statement on the FA website read: "Manchester United’s Nemanja Vidic has had his claim for wrongful dismissal rejected by an Independent Regulatory Commission.
"Therefore, the player's three-match suspension commences with immediate effect."
