Tottenham dominated the opening exchanges at their own stadium but could not turn pressure into goals, with Jan Paul van Hecke and Mateus Fernandes both spurning good chances early on.
Everton grew into the game and finished the stronger side, with Kiernan Dewsbury-Hall and Brennan Johnson missing presentable opportunities either side of a smart late save from Jordan Pickford to deny substitute Lucas Bergvall, as the goalless draw left Spurs without a win or a goal in any of their opening four league matches of the season for the first time in the club's history.
Roberto De Zerbi pointed to a chaotic pre-season, disrupted by World Cup call-ups and injuries, as the root cause of his side's struggles.
"We started this season late in many, many reasons and too late. Many players at the World Cup, many injuries, many problems in Australia," De Zerbi said.
"We brought two players late, Savio and Marmoush, and I started to work with this whole group not last Thursday but two Thursdays ago. So, I think we have not the right physical condition yet."
The Italian insisted his side's underlying performances were better than the results suggested.
"I watch many, many games where the winning team created less than us in the second half. What is the problem? The problem is the goal," he said. "There is a word in Italian, sbloccarsi, that translates as if we unblock ourselves. I think the team is extremely strong, and we can score goals, and we can win games."
David Moyes, whose side extended their unbeaten start to the campaign, admitted he was not satisfied with the performance despite the useful point.
"Was I happy with the performance? Not really, no. I wanted much more from the team, but I told them about coming here - we've not had a great record here," Moyes said. "Tottenham were so good for 20 minutes. They put us under so much pressure and we found it difficult to escape from it. But look, we were worthy, we made a couple of chances, but we didn't play well enough today when we had the ball."
The Everton boss did, however, praise his side's defensive resilience and the clean sheet earned by goalkeeper Pickford and his back four.
"The clean sheet was huge for the team and for everybody. We have a terrific goalkeeper, a back four that does well," Moyes said. "There'll be some games where we might have to play that way just to get ourselves through and pick up points. It wasn't the plan. We were forced into it by Tottenham. I thought we did a really good job of being resilient, hard to score against."