The Athletics won three of four from the Mariners to open the season.
Anderson (1-0) allowed three hits and struck out four in a 97-pitch effort. Seattle's only runs came in the ninth off Jerry Blevins.
Barton singled in a run in the fourth to make it 2-0, had another run-scoring single in the sixth and drove home two more with a base hit in the eighth.
Doug Fister (0-1) took the loss, surrendering six hits and two runs with three walks in four innings.
Seattle's Ichiro Suzuki went 0-for-4, ending a 23-game hit streak against Oakland. It had been the longest active streak in the American League for a player against a particular team.