Rios' sixth-inning blast with Alejandro De Aza aboard snapped a 1-1 tie and put the White Sox on the path to a series-evening win in their three-game set against the Mariners.
Reed closed the door with a pair of strikeouts in the ninth, earning his third save in three Chicago wins to begin the season.
Donnie Veal (1-0) earned the decision with an inning of scoreless relief.
Seattle's Felix Hernandez (1-1) was tagged with the loss after surrendering four runs on six hits over 6 1/3 innings.
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