The reds endured a late spell of Everton pressure to stretch their record unbeaten run in Merseyside derbies to 17 games after the 0-0 draw at Goodison Park.
The draw consolidates their hold on third place in the Premier League with manager Jürgen Klopp eying up Manchester United’s second slot, just one point away.
Liverpool fielded a dramatically different team to the one which beat City 3-nil on Wednesay night.
Mo Salah and Roberto Firmino were notably absent from the starting lineup, which included five changes.
Despite this Klopp's side still had the better of the game with Everton goalkeeper Jordan Pickford pulling off impressive first-half saves from Solanke, Milner and Van Dyke.
Liverpool goalkeeper Loris Karius was on fine form, making an impressive save from a long curling shot from Yannick Bolasie, although Everton seldom showed the forward momentum to threaten his goal.
Everton’s Wayne Rooney was substituted for the match-fit Idrissa Gueye early in the second half.
Yet Liverpool remained firmly in charge, bringing on Alex Oxlade-Chamberlain and Firmino, until a frantic final 10 minutes in which Seamus Coleman and Dominic Calvert-Lewin went close to grabbing an unlikely winner for the home side.