An open game saw Liverpool create another big chance after Salah played in Xherdan Shaqiri, but his shot was saved at the near post by Pickford, before Theo Walcott rounded Alisson at the other end but the ball ricocheted wide.
"The game was what we expected, unbelievably tense, unbelievably hard", said Liverpool manager Juergen Klopp, who rushed on to the field to hug goalkeeper Alisson after the goal.
Liverpool boss Jurgen Klopp described the way he celebrated Divock Origi's dramatic victor as "not cool" following the 1-0 Merseyside derby victory over Everton at Anfield.
Guardiola's side now top the English top-flight league table this term, two points ahead of second place, Liverpool.
His insane celebration, when he ran 40 yards onto the pitch before jumping for joy with goalkeeper Alisson Becker, was not simply because Liverpool had beaten Everton. To see players - and in this case managers - behave with such unbridled exuberance can never be a bad thing and only serves to remind fans that the people they pay to see perform can be just as passionate as they are. "I think this season we are going to do something", he concluded.
Pickford went up for it but fluffed his save and the ball fell for Origi, who came on late for his first Premier League appearance since August 2017, to nod into the empty net.
Origi went close, and should have scored, when he hit the crossbar in the 90th minute, but he then struck the crucial blow by heading in from close range after Jordan Pickford had failed to deal with an awkward high ball from a mishit by Virgil van Dijk.
Arsenal's Pierre Emerick Aubameyang celebrates his goal in the 4-2 win over Tottenham Hotspur at the Emirates Stadium in London
"To be honest I didn't see it (Klopp's run)". But we got it and we will take it.
Speaking after the match, he said, "I think it is the Everton luck when we come to Anfield".
'What can I say about it?
"Man City had the benefit of playing first this week and we knew they had won, so we didn't want them getting too far in front".
The chances kept coming for Everton as Liverpool struggled to find their rhythm, with Andre Gomes denied by a brilliant save from Allison.
'A lot of teams also play counter-attack against us.
Egypt global Mohamed Salah said that he wants to win trophies with Liverpool this year rather than concentrating on individual glory.