The Senegal international was shown a straight red card for a high studs challenge that caught City's Brazilian goalkeeper Ederson on the head in the 37th minute.
He will now miss Saturday's Premier League match against Burnley and two games at Leicester City, one in the league and one in the League Cup.
Mane will, however, be available for the Champions League tie at home to Sevilla on Wednesday.
Liverpool had not appealed the dismissal but argued the ban was too harsh for the incident.
The incident had divided observers and pundits, with Liverpool manager Juergen Klopp saying Mane had not deserved to be sent off.
Ederson was back in training on Tuesday, wearing protective headgear, and named in City's travelling squad for their opening Champions League Group F game against Dutch side Feyenoord in Rotterdam.
The goalkeeper was carried off on a stretcher after Saturday's clash and required eight stitches.
(Reporting by Alan Baldwin/Aditi Prakash in Bengaluru; Editing by Amlan Chakraborty and Toby Davis)
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