A Sergio Aguero’s brace, with a goal in each half and and another goal by Raheem Sterling gave the champions all the three points.
But the hosts also had something to cheer: Harry Wilson’s superb free-kick in the 45th minute, which gave them a goal and a spot on the score sheet.
The victory was City’s ninth consecutive Premier League win over Bournemouth.
They now climb to second on the table behind leaders Liverpool, who beat Arsenal by the same scoreline at Anfield on Saturday.
Sergio Aguero also reached a career landmark of 400 goals for both club and his country Argentina.