Mane was dismissed in the 37th minute after catching City goalkeeper Ederson in the face with a high foot.
He will now miss Premier League matches against Burnley and Leicester, as well as a trip to the latter in the EFL Cup.
The excessive punishment claim was rejected following an independent regulatory commission hearing.
Liverpool manager Jurgen Klopp said on Saturday that neither he nor City counterpart Pep Guardiola thought Mane should have been sent off.
Match of the Day pundits Alan Shearer, Gary Lineker and Ian Wright all said they would have gone for the ball, while Chris Sutton told Match of the Day 2 that Mane’s red card should be rescinded.
However, Shearer said it was still a red card under the current rules. Former England internationals Jamie Carragher and Phil Neville were also among those to say referee Jon Moss was right to dismiss Mane.
Ederson trained on Tuesday as City prepare for their opening Champions League group game against Feyenoord on Wednesday.