The Royals scored three runs in the top of the frame to tie it up 5-5, with the last two coming on a single by Eric Hosmer.
But the Mariners came right back in the next half-inning when Mike Carp led off with a double off Jose Mijares (2-2) and was replaced on the base paths by Kawasaki.
The next batter, Trayvon Robinson, laid down a sacrifice bunt to the left of the mound. Mijares fielded it but threw wildly to first, allowing Kawasaki to come home.
Casper Wells later added an RBI single off Greg Holland for a two-run lead, which proved decisive when Kansas City's Alcides Escobar got a run back in the ninth with an RBI single off Tom Wilhelmsen.
The Seattle closer, however, then struck out Lorenzo Cain with a runner at second to end the game and rack up his 14th save.
Carp finished with three hits and Kyle Seager went 3-for-4 with an RBI as the Mariners finished a four-game sweep of the Royals.
Escobar drove in three runs for Kansas City.