West Brom manager Roy Hodgson felt Peter Odemwingie was in the wrong after the striker exchanged heated words with goalkeeper Ben Foster as frustrations bubbled over late on.
The hosts were in front when Leon Osman's shot took a wicked deflection off Gareth McAuley for an 18th minute own goal and Victor Anichebe's second half effort sealed the points.
But the talking point of the game from Albion's point of view came towards the end of the game when the Baggies stopper felt the Nigerian should have picked up his marker from the short corner and gave him an earful which resulted in a strong exchange of words.
But Hodgson immediately backed his No.1, telling Sky Sports: "We made sure before the game that it was his job (Odemwingie) that he went out when there was a short corner and he would be joined by Billy Jones and he didn't do the job and he was told that he wasn't doing the job but that's one of those things. That was emotional.
"To be fair Odemwingie worked very hard during the game and sometimes when people tell you even if they're telling you things that are not the truth and things you should be able to take as criticism, sometimes you get a bit upset.
"It's an unfortunate thing and I don't like seeing these things but it's all been quickly sorted out and it all just shows that the players still care a lot and we do still want to finish the season in the right way."
And on the game itself, Hodgson felt it was always going to be a tough test against an in-form Everton side at Goodison Park.
He said: "To be fair, we knew it was going to be hard to get points up here today, but there were long period in the game when I thought we were going to be capable of it, until the second went it and that tended to pull the shutter down on our chances.
"But, there are seven games to go and if we keep working hard, we're getting players back for injury as well, I should have quite a few players to choose from and so we have got to hope that we get a winning eleven on the field in some of those last seven games."