For the second game running it was long-serving midfielder Leon Osman who grabbed the vital goal, this time securing all three points for his side after equalising against Norwich at the weekend.
Toffees boss Moyes said: "The last few games Leon Osman has come back to the sort of form we know. He got us a goal on Saturday, he got us another one tonight and he's been a really important member of the squad."
He added: "He's got an intelligence that allows us to work with him on the way he plays and what we asked him to do tonight. Because Swansea are a top passing team and a very good team.
"Him and Tony Hibbert, as far as I'm concerned they're part of the furniture and sometimes they get taken for granted a little bit but they're always there in our corner when we need it and we needed him tonight. We needed to find a goal from somewhere and Ozzy was the man."
It was only the third time this season Everton have picked up maximum points at home, and Moyes is hoping the victory can be the start of a better sequence.
"We needed to win but Goodison's been good to us in the years gone past regarding picking up the victories so let's hope tonight we can start to get on a good run here," he said.
Source: PA
Source: PA