City were frustrated by Rafa Benitez's side in the first 30 minutes, struggling to break the Merseysiders down.
After being denied a penalty, Raheem Sterling reacted by opening the scoring two minutes before the interval.
City then doubled their lead with that stunning strike from Rodri early in the second half, before Bernado Silva finished things off when he slid the ball past Jordan Pickford after it deflected into his path.
The game got off to a slow start but City eventually began to find their stride with Silva and Cole Palmer both testing Pickford and Ilkay Gundogan heading onto the bar.
Guardiola was without key men Jack Grealish and Kevin De Bruyne, relying on youngsters Palmer and Phil Foden for creativity.
But it was fellow forward Sterling who thought he had done enough to create a real opening for his side.
City were awarded a penalty when the England international went down on the edge of the area, but referee Stuart Attwell reversed his decision after reviewing the incident on the pitchside monitor.
Sterling was involved again soon after, however, and, after a stunning ball from Joao Cancelo, he couldn't miss to put City ahead.
Everton cleared from an attack, but only as far as Cancelo, and the Portuguese curled a brilliant ball back into the box with the outside of his boot for Sterling to sweep in.
City then doubled their lead through Rodri's bullet into the top corner, which all-but killed the contest off.
The Spaniard lashed into the top corner from 20 yards following a poor clearance by Allan - talkSPORT's Micky Gray describing Rodri's strike as something 'out of the top drawer'.
"What a strike from Rodri, absolutely out of the top drawer," he said.
"The ball falls to him 25-yard out, Fabian Delph tries to close him, but he gets his laces through it, what a super strike, [Jordan] Pickford has no chance."
Silva, who has been one of City's best players this season, made sure the points would remain in Manchester when he poked the ball home after Palmer's strike deflected kindly in his path.
City are now up to second in the league, overtaking Liverpool, and remain three points behind leaders Chelsea.