Judging from the few Mariners games that I have watched on TV the past couple days, they are using either RHodes or Nelson based on the situation and who is at bat. On Sunday, they had Hasegawa, Rhodes and Nelson all pitch to batters in the 9th inning based on matchups. Since Nelson is the rightie, he will probably get most of the saves while Rhodes would normally be used as a situational lefty, depending on who the opposing team has coming up. Apparently, under certain situations, they said that Hasegawa could also get save chances.
I highly doub that Sasaki's job is in any jeopardy long term though. He just needs to hire someone to carry his suitcases for him.