Scottish striker Steven Naismith bagged the Toffees the lead as early as the 12th minute but they failed to find a second and were punished by Sebastien Bassong's last-minute equaliser.
Moyes said afterwards: "That was pretty hard to take. But they were always in the game while it was only 1-0, that is the nature of the Premier League.
"I thought we played well for 25-30 minutes. We started well and controlled the game.
"Norwich grew into the game, give them credit, they had one or two opportunities.
"I thought we had got through the spell and we probably looked more likely to make it 2-0 than for them to equalise. But they did and that was it. We came away with a point."
John Heitinga and Tim Howard were arguably to blame for the late leveller as Javier Garrido's free-kick was converted by Bassong.
When asked who was at fault, Moyes added: "A little bit of both. I think the ball was in the air long enough and also I don't think it was a free-kick.
"Today was always going to be a tough game. Norwich have played well and have had some good results. With what we had available to us it was always going to make it quite hard.
"It was going to be tight - I couldn't see us winning it that comfortably. We were chasing it and we didn't have an awful lot to chase it with. They are the facts."