Moyes praised matchwinner Baines after the left-back took his tally to the season to seven, and revealed the defender has been playing through the pain of an ankle injury.
“Leighton keeps getting better when teams try to stop him as he works on how to alter doing things,” said the Goodison manager. “He is a great player.
“He is great for a manager as he wants to get better himself, we are very fortunate to have him here.
“Bainesy has had a bit of a problem with his ankle. He has played through with an ankle problem which is a credit to him. He has the determination and he is turning out to be as good a left-back as there is.
“He is carrying his ankle a bit but is missing no training. He is a great player to have. He is a really good footballer, intelligent.
“I don’t know I like his style of hair-dos but apart from that I think he is all right.”
A John Heitinga error allowed West Brom substitute Shane Long to pull a goal back midway through the second half and set up a nervy finale.
And Moyes said: “It was harder than I liked it to have been. We were worthy to be in front and we needed the second goal before the break.
“We gave away a poor goal but apart from that not done a lot wrong all night.
“Games in the Premier League are tight. You have to earn it and we hard to earn it by doing the dirty things right.”
Moyes, who saw Darron Gibson make his return from injury with a second-half substitute appearance, added: “We have had a decent January. I don’t know what other teams have had but we have picked up draws and I think sometimes you just have to do that.
“We’ve just stuck at it. We have not been as fluent as we would like to be at times. But you look at the results in the Premier League and there is very little between the top and the bottom. There is nobody winning games at a canter.”