Asked if Sunday was one of his worst days on the job, the downcast manager told reporters: “Yes. Of course, it is disappointing.”
The result sees City move level on points with Arsenal in second place, while United remain in sixth.
Players are expected to wear black armbands, with wreaths to be laid on the Old Trafford pitch and a minute’s silence observed before kick-off.
“Sir Bobby will always be remembered as a giant of the game,” the club added.
Salah added another goal in stoppage time to seal the victory for Liverpool.
Rooney, the former Manchester United and England striker, was appointed Birmingham manager on Wednesday.
Despite dominating 78 ball possession, United could not salvage a point against Palace on their home turf.
United have several injury issues to deal with and have confirmed Sergio Reguilon will be absent against Crystal Palace on Saturday.
Meanwhile, Chelsea are also into the fourth round after the Blues pipped Brighton one-nil at Stamford Bridge.
Jonny Evans, who was making his first United start after eight and half years, provided Fernandez with the assist for the goal from the middle of the pitch.
