Manchester City's 1-0 win at Bournemouth a day earlier moved the defending champions back to the summit, meaning Liverpool had to win at Goodison Park on Sunday to go top once more.
But with Mohamed Salah wasting the clearest chance of the game for Liverpool, Everton prevented their rivals from taking another step towards the club's first title of the Premier League era.
After a high-octane start, Salah fired the game's first shot in the 15th minute but Jordan Pickford, whose error handed Liverpool victory in December, made a comfortable save.
Salah should have given Liverpool the lead when played through by Fabinho, but the Egypt star could not find the right angle to beat Pickford as the Everton goalkeeper made himself big.
After the interval a rare loose pass from Fabinho led to an Everton corner but Alisson held on to Dominic Calvert-Lewin's downward header.
Michael Keane made a superb challenge to deny Salah another shot on Pickford's goal before Klopp turned to Roberto Firmino and vice-captain James Milner from the bench.
Bernard was close to turning in a fizzed low cross from Richarlison as the game became increasingly stretched in the second half, before an unmarked Cenk Tosun mishit his shot from Gylfi Sigurdsson's free-kick.
December's derby saw incredible late drama as Pickford's mistake allowed Divock Origi to score a 96th-minute winner, but Everton held firm in the 200th top-flight meeting between the clubs for a result that may prove pivotal in the title race.
What does it mean? Advantage City in title race
New leaders City have won five Premier League games in a row to keep the pressure on Liverpool, with successive 1-0 victories ground out by Pep Guardiola's men.
With only nine games to go, Liverpool will now have to play catch-up, as City next host Watford, who lost 5-0 at Anfield earlier this week.
The Reds have also now failed to score in three of their past four games in all competitions.
Pickford makes amends
Everton were left devastated after Pickford's mistake in added time at Anfield handed Liverpool a priceless victory to boost the Reds' title challenge in December.
But England's number one was on top form to keep Liverpool at bay in the return game, with a huge save from Salah in the first half ensuring his side kept a determined clean sheet.
Salah still hunting 50th goal
A run of three games without a goal would ordinarily not be considered a concerning issue, but Liverpool talisman Salah lacked his usual surety when presented with chances. A one-on-one with Pickford midway through the first half was the clearest opportunity either side had to break the deadlock.
Salah is one away from his 50th Premier League goal for Liverpool and, as the Reds begin to feel the heat in the title race, he is perhaps beginning to be affected by the growing pressure.
Key Opta facts
- Liverpool have failed to score in two of their last three Premier League games, having failed to find the net in just one of their previous 27 in the competition.
- Liverpool didn’t record a single shot on target after Trent Alexander-Arnold’s effort in the 54th minute of this match.
- 67 per cent of Everton’s three shots on target in this match came via their substitutes (one for Richarlison, one for Cenk Tosun).
- Mohamed Salah has gone three Premier League games without a goal for the first time as a Liverpool player.
- Liverpool’s Alisson has kept 17 clean sheets in the Premier League this season – the most by a keeper in their debut season in the competition since Jose Reina in 2005-06 (20).
34 - This was the 34th goalless draw between Everton and Liverpool in English league football - 15 more nil-nils than any other fixture in the history of the Football League. Anticlimax. pic.twitter.com/sE4D74thYe— OptaJoe (@OptaJoe) March 3, 2019