For a long time it appeared Paul Lambert's side would be celebrating their second surprise success on Merseyside, having beaten Liverpool in December.
But Marouane Fellaini scored his second of the afternoon in injury time to add to Victor Anichebe's first-half strike and cancel out Christian Benteke's double and a goal from Gabriel Agbonlahor.
Everton's remarkable powers of recovery came to the fore again and they are second only to Manchester United having taken 21 points from losing positions this season, although the draw was not enough to take them back into the top four for the first time since December 22.
Centre-back John Heitinga had a particularly poor afternoon and he was replaced in a double substitution along with Kevin Mirallas.
There were a smattering of boos from the Goodison Park crowd when the Belgium forward was taken off but the jeers which rang out when Heitinga left the pitch were directed at the player rather than the manager.
"I think they were not happy with my decisions," said Moyes of the crowd reaction. "I think they were the same at Bolton (in the FA Cup last weekend) as well, disappointed with the decisions. I'd rather they direct it at myself if that is the case but we will keep going and trying to earn those victories."
Of the weaknesses at the back, the Scot added: "I'd like to say I didn't think we should be in that position (3-1 down) but we were because we defended poorly - the defending let us down. If anyone doesn't think Benteke, (Charles) N'Zogbia and Agbonlahor can't cause you problems then they're wrong and you have to be ready for that.
"We were ready but we didn't just deal with it that well. I had a strange feeling we might have to score four goals to win it. I hoped I was going to be wrong but it was the way it panned out but it wasn't for the lack of effort from the players.
"With the week we've had - Bolton away in the cup, at home to West Brom and then Aston Villa - if at the start of the week you'd given me a win in the cup and four points I might have taken it."
- Sports & Recreation
- Christian Benteke
- John Heitinga
- Manchester United
- Gabriel Agbonlahor
- Marouane Fellaini
- Victor Anichebe