The Everton manager rues his side's profligacy as they squander a 1-0 lead in the dying moments, also disputing the award of the free kick in the build-up to the equaliser
An early Steven Naismith goal put Everton ahead but, despite many chances, they were unable to capitalise on their dominance and the Canaries' centre-back made them pay.
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Moyes was clearly frustrated with his side's wastefulness, also briefly directing his ire at the referee for a decision in the build-up to the equaliser.
"It's hard to take, but that is the nature of the Premier League - Norwich were always in the game, even at 1-0," Moyes told reporters.
"I thought we were more likely to go on and make it 2-0 rather than than them equalising. But they did and we only came away with a point. Was it a free kick? No."
The long-serving Everton manager was, however, keen to praise the form of goalscorer Naismith, who notched his third goal in three Premier League starts for the side.
"I hoped that he would come here and get us goals. If he can finish the season with seven or eight, then that will be a good return from him," he continued.
"He is now beginning to get them so good on him."