Kazmir (6-4) surrendered a fifth-inning single to Justin Smoak to go with two walks and seven strikeouts, in extending his unbeaten streak to eight starts.
Bourn's bases-loaded blast in the fifth inning -- the first grand slam of his career -- highlighted a 13-hit attack for Cleveland, which avoided a three-game sweep.
Asdrubal Cabrera went 2-for-4 with a solo homer, a two-run double and four runs scored while Carlos Santana added three hits, including a pair RBI doubles for Cleveland -- which came into the contest trailing American League Central-leading Detroit by 3 1/2 games.
Seattle starter Joe Saunders (9-9) was roughed up for six runs (five earned) on nine hits in a 4 2/3-inning effort.
The loss snapped an eight-game Mariners' winning streak.