The reigning champions headed into Saturday's Premier League clash at Goodison Park eight points behind Liverpool, who beat Sheffield United 1-0 in the day's early kick-off.
Dominic Calvert-Lewin cancelled out Gabriel Jesus' opener before half-time, but Riyad Mahrez caught out Jordan Pickford from a free-kick 71 minutes in and Raheem Sterling made sure of the points.
With the gap at the top back to five points after seven matches, Guardiola acknowledged the pressure is on his side to keep pace with last term's runners-up.
"We knew Liverpool won again and the gap was eight points, so big," Guardiola said at his post-match news conference.
"So, we won three points and it's five [points] in September, with a lot of points to play for.
"We know which team we're playing right now - they are seven victories in a row, champions of Europe and playing good football.
"This is our rival and we know we can't drop too many points because last season they lost just one game.
"The season is long, we have to keep going and be there, maybe we can push them a little bit more and try to make the third Premier League in a row."
Ederson produced a couple of impressive second-half stops to help City on their way to a fourth win on the bounce in all competitions, much to the delight of Guardiola.
"He saved two incredible chances at 2-1 and it was a good game of football," he added.
"We were absolutely outstanding. At Goodison Park it's always difficult - we started incredibly well as usual, we created chances and scored.
"After that we had a lot of problems to control out left side with Gylfi Sigurdsson, they had one more player there, we had a lot of problems.
"In the second half, except set-pieces and conceding free-kicks and long balls we played good."
Everton winger Theo Walcott was taken from the field on a stretcher in the opening stages of the match after blocking Sterling's cross with his face.
Walcott was taken to hospital but Marco Silva confirmed after the game that the 30-year-old had been discharged.
"He's already in our dressing room," Silva told reporters. "He's okay, he's taking his shower and will go home to rest.
"He had a scan [at the hospital]. Up to now everything is okay. We're just waiting to speak more to the doctor."