Home fans spilled onto the pitch moments after Phil Jagielka's missed penalty had given the Bees one of the biggest results in their history which came courtesy of a 4-3 shoot-out win after the scores were level at 1-1 after extra time.
Moyes was disappointed with the fan's purported action, and claims he had to hold himself back from tackling the person he thinks was responsible.
The Everton boss said: "I thought it was terrible. I saw a fan throw a bottle into our supporters. I said to him 'I saw your face. I saw who you were'. He got a bottle and went right up and threw it right into the middle of our supporters.''
Moyes slammed the decision to hold the penalty shoot-out in front of home supporters.
"I think the Police decided to take the penalty kicks at the end for security reasons. That proved to be a right bloomer, didn't it?'' he said.
"If the police want me to follow it up and try to find the individual that threw the bottle into the crowd then that would be fine by me, it wouldn't be a problem. I didn't grab him. I wanted to but obviously that would have brought more attention to the situation.''