The Toffees entered the match on the back of three straight draws, with Frank Lampard's men yet to discover the winning formula.
Liverpool, on the other hand, were looking for their third straight victory after thrashing Bournemouth and edging past Newcastle on Wednesday.
The Reds looked dangerous and hit the woodwork three times during the match. Everton were also denied once but both sides had to settle for a point from a sometimes feisty goalless Merseyside derby.
Tom Davies found the outside of a post for Everton midway through the first half before Jordan Pickford pushed a Darwin Nunez shot on to the crossbar and Luis Diaz hit the frame with a follow-up effort.
Then in stoppage time the outstanding Pickford made a superb save to push Mo Salah's shot on to the upright.
Conor Coady had the ball in the net in the 70th minute but when his goal was chalked off by VAR for an offside there were ugly scenes as a bottle appeared to be thrown, narrowly missing Jurgen Klopp, while another fan ran on to the pitch from the Gwladys Street End to confront referee Anthony Taylor.
But the day belonged to the goalkeepers, with both Pickford and Alisson Becker making fine saves in the second half.
The result kept Everton winless to start the Premier League season though they were probably happier with the result than Liverpool, who are still yet to win away this campaign and have already dropped nine points from their opening six games.