The Toffees had the better of the chances at Goodison Park and the Baggies were indebted to some fine goalkeeping from Ben Foster to keep the scores level.
Olsson told the club's official website: "A point at Everton is a good one. They probably had more clear-cut chances than we did, but we dug in.
"We spoke before the game about getting on with the season and getting some points on the board.
"Our shape in the first two games has been good and that should offer us some stability."
West Brom have taken one point from their opening two Premier League fixtures this season.