Although the fee for the 24-year-old Serbia and Montenegro international has yet to be disclosed, Vidic is expected to arrive at Old Trafford early in the New Year.
Ironically, Vidic comes at a time when United centre-back Rio Ferdinand has shaken off a poor spell of form to impress.
After 139 games without a goal for United, Ferdinand broke his duck against Wigan a fortnight ago and he was again on target against West Brom.
Albion came to Old Trafford to frustrate United with a five-man midfield supporting lone striker Nathan Ellington, but the plan went out of the window following a 26th minute injury to defender Paul Robinson.
Robinson, who had been given the job of marking Wayne Rooney, crashed into Thomas Gaardsoe when he attempted to clear the ball in the Albion box.
After a six-minute stoppage for treatment, he was taken off on a stretcher suffering from concussion and was replaced by Diomansy Kamara. Kamara was pushed up front to join Ellington and within two minutes Albion were behind.
Ferdinand was instrumental in the move when he raced on to Ji-Sung Park's pass down the right.
The England defender's low cross into the box hit Darren Carter and rebounded to Park, who touched it on for Paul Scholes to score with a crisp low left-foot shot from 15 yards.
United made it 2-0 two minutes into first-half injury-time.
Ryan Giggs clipped his corner into the centre of the Albion penalty area and Curtis Davis appeared to slip as Ferdinand rose majestically to head the ball into the top left corner of the net.
The game was effectively over with United posting their third goal in the 63rd minute.
Park's flick sent substitute Alan Smith racing down the right and his cross into the centre was headed home by Ruud van Nistelrooy eight yards out.
Albion's only real chance of the game came three minutes from time when former United player Ronnie Wallwork hit a weak close-range shot straight at goalkeeper Edwin Van der Sar.