The Toffees were twice forced to come from behind in the match, with Marouane Fellaini's equaliser snatching them a share of the spoils in the 92nd minute.
Moyes hailed his side's spirit and never-say-die attitude, but he was unhappy with the way they defended.
"I thought we played well but didn't defend well," said Moyes.
"Great credit to the players as they never gave up and kept going. We found ourselves chasing the game all the time and that makes for frantic football."
The draw leaves Everton fifth in the Premier League table, while Villa remain in 19th place.