Kawasaki's three-run double came in the fifth inning and broke the game open, giving the Mariners a 7-2 lead.
Jaso delivered a run-scoring single the third inning that put the Mariners ahead for good at 2-1 and Kyle Seager belted a solo homer as Seattle claimed the opener of a three-game series.
Jason Vargas (7-4) allowed four runs on nine hits over 5 2/3 innings while recording eight strikeouts for Seattle.
Angels starter Ervin Santana (2-7) issued six walks over 4 2/3 innings. The right-hander was lit up for seven runs on eight hits.
Kendrys Morales socked a pair of solo homers and Mike Trout went 4-for-4 with two RBI for Los Angeles.