The Toffees took the lead when Bryan Oviedo capitalised on Steven Whittaker’s mistake and teed up Steven Naismith, who emphatically fired past Norwich goalkeeper John Ruddy.
Grant Holt came close in the 24th minute when he beat three defenders before firing over the crossbar, while John Heitinga cleared off the line after Robert Snodgrass has beaten Tim Howard in the second half.
With Everton seconds from victory, a Norwich free-kick led to Sébastien Bassong heading past the Toffees goalkeeper to snatch a point for the Canaries and extend their unbeaten run to six games in the Premier League.
“I thought Steve Morison mis-kicked the ball in front of us and I don’t think it was a free-kick,” Moyes told BBC Sport.
“It was a tough game but we should have seen it out and I am disappointed we didn’t.
“We had the opportunities to make it 2-0 but Norwich were always knocking on the door.”
Meanwhile, Norwich winger Anthony Pilkington revealed his delight at securing an invaluable point on Merseyside.
Pilkington posted on his Twitter page: “Great point, always a tough place to come!! Congratulations to Sébastien Bassong on his goal! Grant Holt tried to rob it!”