Bernardo Silva's opener for City was cancelled out by Iliman Ndiaye for Everton, leaving the hosts with one win in the past 13 games.
Dyche said afterwards: "I was worried about coming here today because it will change, at some point they will hurt someone, but I felt we handled it well - the underlying feeling that they will pull a result out of the bag at some point.
"Great energy from the players. We saved the penalty (from Erling Haaland) and Jordan (Pickford) and the analysis team deserve credit.
"Jordan made a lot of good decisions today.
"There's no easy run in the Premier League and you have to be committed to the cause for all of them.
"I am pleased overall with the performance and getting a point."
On the transfer window, he added: "We still have a PSR, no PSR problem but we still have the rules and regulations. It has been a tough process since I've been at the club, selling more than I buy. It has been a touch challenge."