Goals from both full-backs Greg Cunningham and Javan Vidal on top of striker David Ball’s pair secured the points in the wake of last week’s promising opening win at Hull.
Everton had experienced Nigeria striker Yakubu leading the attack as he returned from injury, but it was City who presented the early threat up front with the lively Ball.
The striker has started the season in fine form, and he showed superb composure after 12 minutes when he raced through to lob advancing goalkeeper Iain Turner to put City in front.
The Blues might have increased their lead minutes later with a smart move that saw Alex Nimely play in Paul Marshall only for the midfielder to tuck his shot inches past the post.
No matter - City stretched their advantage after 16 minutes with a delightful solo effort from left-back Cunningham, who swept past two tackles then fired an angled shot past Turner.
Yakubu, whose campaign was wrecked last season by a ruptured achilles in November, understandably looked rusty but Everton snatched a goal back after 27 minutes when Adam Forshaw spun and shot past Gunnar Neilsen.
Marshall delivered an audacious long-range free kick that Turner held right on his line just before the break to draw applause from a crowd of 369 on a surprisingly chilly night.
Half-time substitute James Poole wasted no time in letting Turner know he was around with a fierce shot that the keeper turned onto the outside of a post for a fruitless corner.
But Poole responded by setting up Javan Vidal for City’s third goal after 54 minutes, with the raiding right-back showing admirable finishing ability to leave Turner helpless.
Eager Ball still had time to show his clinical finishing power when he left centre-half James McCarten tumbling in his wake to coolly beat Turner for the fourth after 85 minutes.
The win would have been more emphatic but Alex Nimely, who played well despite some tough treatment, hit an 80th-minute penalty too close to Turner after the keeper had brought down Cunningham.
