Jurgen Klopp's Liverpool side went into the game, their first competitive match in 103 days, knowing a victory would allow them to secure their first league title in 30 years on Wednesday with a win at home to Crystal Palace.
It was more of the same in the second half, with Liverpool dominating possession and going close on several occasions but continuing to lack a cutting edge on their finishing.
"It was very intense," said Liverpool defender Virgil van Dijk. "Both sides created small opportunities, they had the big one against the post. Everyone is just happy to be back, we didn't lose and it is another point towards our goal."
With 10 minutes remaining Dominic Calvert-Lewin's clever flick from a low Gylfi Sigurdsson cross was pushed out by Liverpool keeper Allison and Tom Davies' follow-up shot hit the post.
Liverpool, who left Mohamed Salah on the bench, lacked their usual potency in attack with their best opening coming in the 35th minute when Roberto Firmino screwed a shot wide from a promising position while Fabinho went close late on with a curling free kick.
Both Klopp and his Everton counterpart Carlo Ancelotti handed players first league starts with Japan's Takumi Minamino in the Liverpool attack and 19-year-old Anthony Gordon up-front for the home side -- although neither were able to make much impact.
"We didn't have enough chances. Most of the time we were dominant but they had the biggest chance," said Klopp. "I liked a lot of things about our game but we didn't have those moments in attack, we didn't have the rhythm.
"Everton defended really well and we were not smart enough to use the space. That is normal when you start to play again.
"It was a real fight. Both teams showed they understand it is a derby. Intense, physical, all players were all in.
"The point is one we deserve. When I think of all the derbies here, they looked pretty similar. Everton were well organised so we had to run a lot but we looked fit and ready."
Information from Reuters was used in this report.