Raul Meireles hit the opener for Liverpool, but Sylvain Distin and Jermaine Beckford turned the game on its head shortly after half-time. Dirk Kuyt salvaged a point from the penalty spot when goalkeeper Tim Howard fouled Maxi Rodriguez.
Moyes admitted some of his players were overawed early on in their first derby at Anfield but felt they got to grips with things for the start of the second half.
"I am disappointed we didn't take all three points after the second half but delighted to only come in 1-0 down after the first half because Liverpool were the better team," he said.
"In the first half we never got to the pitch of the game and it looked as if we had some young players in the team who looked overawed by the start of the game.
"Liverpool won all the duels so it needed a word to correct it at half-time and in the second half we were terrific.
"I thought we had a chance when it went to 2-1 because we were on top but we were missing three or four big players.