Foden's injury occurred after just 22 minutes of play and the England midfielder was unhappy when he was forced to leave the field.
After the sending off, Man United created the most dangerous chances. Marcus Rashford hit the post with a free kick and Kobbie Mainoo also hit the woodwork with a shot from distance. But there were no goals for United.
The decisive goal instead came from the away team, City, through superstar Erling Braut Haaland with half an hour left.
Arsenal and Man City share the lead in the league. United are in 13th place.
Earlier on Sunday, Brighton beat Coventry 5-0, with Swedish national team midfielder Yasin Ayari scoring one of the goals.
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