The Serbia international was forced to come off just before half-time at Old Trafford after colliding with United goalkeeper David de Gea.
Vidic was left dazed after being accidentally hit in the head during the collision and was admitted to hospital for examination but he has now returned home.
"Nemanja Vidic has been released from hospital after suffering concussion," said a Manchester United statement.
Vidic is not due to play again until 24 November, when David Moyes' side visit Cardiff in the Premier League.
But the 32-year-old will have to prove to the club's medical staff that he is fit to play again and no timescale has been put on his return to training.
Concerns over head injuries have increased recently in the wake of Tottenham's decision to allow keeper Hugo Lloris to play on despite suffering concussion after colliding with Everton striker Romelu Lukaku.
Lloris was subsequently rested for Spurs' game with Newcastle at the weekend after failing to successfully pass a computerised concussion evaluation system.
Play also had to be held up at Old Trafford for a second time on Sunday, after Arsenal goalkeeper Wojciech Szczesny was injured following a clash of heads with United's Phil Jones.
Headway, the head injury charity, had highly critical of Spurs but said it was too early to say whether Arsenal had made the right call on Szczesny.
Headway spokesman Luke Griggs said: "I don't want to get involved in judging the decision of Szczesny - every situation is unique and different and the medical team have made a call.
"With Vidic, they have taken him off after the medical team deemed him not fit to continue. He was unsteady on his feet, there were more obvious signs.
"From the very start the whole situation was to raise awareness about head injuries.
"It has made people think. Especially the impact it has on youth and amateur football, about taking risks when they are playing football. They might want to be a hero and carry on - and they don't have medical teams at amateur football.
"If they are unsure and unsteady on their feet, it is best to go off."