A subdued Everton finally conjured up some pressure in the last 15 minutes, with Cenk Tosun and Dominic Calvert-Lewin both missing chances.
Coleman told evertonfc.com: “We felt Liverpool would be happier than us with a point but we started slowly and did not get to grips with the game.
“But in the second half we got tighter, pressed people more and played on the front foot.
“We are disappointed not to have come away with more after the chances we had at the end. Dominic is disappointed but he will work on that and I could have maybe got my toe on one at the end.
“It was a clean sheet against a good Liverpool side, though, and to get the fans behind us we need to play from the first minute. They react to us and if we are getting in amongst people you can see how positive the atmosphere is.
“Our two centre-halves were very good. Jags (Phil Jagielka) was immense and played a like a captain and leader. We owe it to ourselves and the fans and everyone at the club to finish as high up the table as we can.”