The pivotal moment in a game of few clear chances came just past the hour mark when referee Chris Kavanagh awarded a penalty against Tottenham keeper Hugo Lloris for a foul on Richarlison but changed his mind after checking the pitchside monitor.
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Tottenham substitute Giovani Lo Celso almost won it for the visitors late on with a shot against the post, but a draw was just about a fair result for the two out-of-form sides.
Conte, appointed as manager on Tuesday, will take some satisfaction from Tottenham's first league clean sheet for more than two months as the Londoners drew for the first time this season.
Heading into the international break Tottenham are in ninth place with 16 points from 11 games with Everton, who ended a run of three successive league defeats, a point worse off.
After watching his side at Goodison Park nothing would have changed those sentiments, although he would have been impressed with his side's attitude which was far more tenacious than in some of the limp displays under Nuno Espirito Santo.
While Tottenham defended ruggedly at times, their attack functioned only sporadically with striker Harry Kane again looking well below his best as the visitors failed to manage a single attempt on target.
That said, Tottenham did have chances.
Emerson Royal headed over in the first half while seconds before halftime Sergio Reguilon scooped a half volley over the bar after being picked out by Harry Kane.
Everton were the better side after the break and thought they had been awarded a penalty when Richarlison broke into the area and went flying in a collision with Lloris.
Kavanagh immediately pointed to the spot but after consulting with the VAR official via his earpiece, he trotted over to view the incident on a monitor.
The video was inconclusive with Lloris seemingly making minor contact with the ball before taking out the Brazilian striker, but to loud boos Kavanagh cancelled the spot kick.
After that it was a messy affair with tempers fraying and Lo Celso's shot against the post was a rare moment of quality.
Conte again spoke of the many improvements that he needs to make, but said Sunday's showing was a good starting point, even if Tottenham failed to manage a shot on target.
"Today I have seen the heart of my players, the passion, the will to fight, the will to sacrifice, the will to understand that there was a moment very, very difficult for us and to see this type of reaction makes me more confident for the future," the Italain told reporters.
"Sometimes you can teach to our players the tactical aspects and improve your players' physical aspects but the heart, the passion, the will, the intensity, you have or not.
"My players are showing me these aspects and that we are in a good position and it is a good point to start for me."