After a goalless opening period, the Toffees upped the tempo after the interval to claim the lead after 54 minutes with a deflected Osman shot.
The goal was Everton's reward for a spell of dominance and the Merseysiders made the game safe when Leighton Baines drilled in a penalty after a foul on Seamus Coleman by Phil Jones.
"In Leon Osman we had someone who was a match-winner by a long way," said Moyes. "He deserves to be `bigged up', if you want to say that.
"He has had to take on the mantle since we lost Steven Pienaar and with Mikel Arteta out. We have not got a great deal of creativity or that style of player at the moment.
"I thought Osman, when he got it, made things happen and scored a really important goal for us. That got us on our way."
Source: PA
Source: PA