Liverpool made five changes to their line-up and were without the injured Mo Salah and rested Roberto Firmino until he came on late on, but still could have taken all three points.
Alex Oxlade-Chamberlain went close when he came on, before Everton had two huge chances late on in the game, but both sides kept a clean sheet.
That will suit the Reds more than the Blues, with Liverpool now on a 17-game unbeaten derby run going back eight years.
Goalkeeper and Defence
Loris Karius (7): Great save from Yannick Bolasie’s strike from distance. Some good distribution. Other than that, had little to get involved in.
Nathaniel Clyne (7): Played it safe on his first game back, but looked sharp and steady. Kept Bolasie quiet and played a big role in the clean sheet.
Dejan Lovren (7): Dealt with Cenk Tosun well and was able to play the ball out from the back a few times.
Virgil van Dijk (7): Controlled as always and coped well. Took too long on the ball on occasion but generally confident and assured. One free kick went straight to Jordan Pickford.
Ragnar Klavan (7): Theo Walcott did nothing all game and the Estonian did very well in a changed position. Solid and unspectacular.
Midfield and Attack
Gini Wijnaldum (8): Playing in the deepest role in the midfield as a trial for Tuesday night with Jordan Henderson suspended and Emre Can injured, he was Liverpool's best player. Passing accuracy of around 95% and looked comfortable in a slightly new position.
Jordan Henderson (7): Lots of forward passes and crisp balls into the front three. Tired a little towards the end but a decent enough display.
James Milner (7): Offers quality on the ball and energy without it. Rested for Tuesday when he came off, and Liverpool looked a different team without him.
Danny Ings (6): Struggled to get into the game on his first start under Jurgen Klopp. Chased and harried but he does not have the quality of the usual right-sided front player.
Dominic Solanke (5): Should have scored when the chance came and was kept very quiet by Everton's defence. Fortunate to last the whole game.
Sadio Mané (6): Unable to make the most of the ball when he got it. Nowhere near the level he reached on Wednesday against Manchester City, but that is no surprise. Will be in the side again on Tuesday.
Alex Oxlade-Chamberlain (6): Added some urgency to the midfield; shot over the bar once he came on but could not force a winner and faded after a promising first five minutes on the pitch.
Roberto Firmino (5): Stuck out on the left-hand side, he struggled to get involved. Walcott ran past him with ease on a couple of different occasions.
Trent Alexander-Arnold (n/a): Only came on to halt Everton's late pressure.