Raheem Sterling put City ahead just before the break with a fine finish after a fine ball by Joao Cancelo before Rodri rocketed in a stunning second. Bernardo Silva rounded off the scoring with a well-taken third, four minutes from time.
The win was a strong response to Chelsea and Liverpool's wins over Leicester and Arsenal respectively on Saturday, moving back into second before switching their attention to Wednesday's Champions League clash with Paris Saint-Germain.
Pep Guardiola sprang a surprise with his team selection as he handed a league debut to teenage midfielder Cole Palmer, 19, in City's attack. Phil Foden lined up on the left, with Sterling right and Palmer through the middle as a False No.9. Sterling found Foden after 11 minutes but his header narrowly missed the target in the game's first notable chance.
Everton, already struggling with injury absences to key players including Yerry Mina and Dominic Calvert-Lewin, suffered another setback early on when Demarai Gray hobbled off, to be replaced by Alex Iwobi.
Palmer then played a fine through ball for Silva but he was denied by Jordan Pickford and Ilkay Gundogan's follow up bounced off the bar to safety. Everton struggled to leave their own half and after they gave the ball away once too often, Sterling pounced and surged into the box and was nudged off the ball by Michael Keane. Referee Stuart Attwell gave a penalty but after being told to consult the pitchside monitor by VAR Chris Kavanagh, he changed his mind and ruled out his original decision.
Everton thought they had escaped to half time with the game goalless, but Cancelo and City had other ideas. City's full-back floated a sublime ball with the outside of his right foot that evaded the Everton defence and Sterling, on his 300th league appearance, finished first time to complete a wonderful move.
City doubled their lead in some style soon after half time, as Rodri thundered in a powerful drive that left Pickford with no chance. As replays of the goal were shown on the big screen at the Etihad afterwards, the finish drew gasps from the home crowd.
Rafael Benitez's men have won just once in nine matches in all competitions and never looked likely to end that streak against a City side who strolled through the closing stages. There was still time for Silva to slip through undetected and slot past Pickford to round off a fine day for Guardiola's men.