The 26-year-old Argentine, City’s top scorer with 19 goals, went down holding his left leg after tumbling under a challenge from Muhamed Besic in the early stages of the Premier League game at the Etihad Stadium.
After receiving treatment, he was helped to the side of the pitch by one of City’s medical staff, but with tears visible in his eyes, the decision was made to send on 18-year-old Jose Pozo in his place.
The injury to Aguero falls at a particularly unfortunate time for City manager Manuel Pellegrini, whose side face a crucial Champions League group game away to Roma on Wednesday.
The defending Premier League champions, who began the weekend in second place in the table, were also given extra incentive to beat Everton by leaders Chelsea’s 2-1 defeat at Newcastle United earlier on Saturday.
Injuries restricted Aguero to only 23 league appearances last season, but he has been injury-free and in scintillating form since the start of the current campaign.
The former Atletico Madrid striker had scored 16 goals in his 14 previous City appearances, including a hat-trick against Bayern Munich in the Champions League and all four goals in a 4-1 league win over Tottenham Hotspur.