Romelu Lukaku gave Everton the lead by finishing off Kevin Mirallas' cutback before the provider turned scorer to add a second immediately after the interval.
Teenage midfielder Tom Davies scored the third by finishing off a mazy solo run before Everton debutant Ademola Lookman added further gloss to the scoreline by slotting through Claudio Bravo's legs in injury time.
Koeman felt the timing of his side's goals was particularly important as Everton strengthened their hold on seventh place in the Premier League.
"You cannot plan a game like this," Koeman told BBC Sport. "It is always difficult against City, they play great football but the second half was perfect in every aspect.
"We scored at the right time in the first half and then to score straight after half-time made it very difficult for them.
"I think it is a big compliment to Everton - the organisation defensively. It makes the final result and the way we played perfect."
Koeman defended under-fire Pep Guardiola, who suffered the heaviest league defeat of his managerial career, and backed the City manager to succeed.
"Pep Guardiola knows it is a project at Manchester City," Koeman said.
"Of course maybe they expected better results and a defeat like this is really strong, but Pep has the experience to turn it around. I don't doubt it."