Suzuki drew a leadoff walk from Yoervis Medina (1-2) starting the ninth and went to second on a Jayson Nix sacrifice.
After Reid Brignac flew out, Stewart delivered a single to left field, chasing home the decisive run.
The late tally made a winner of David Robinson (4-1), who tossed a scoreless eighth inning after Yankees starter David Phelps had yielded just one run on three hits over six frames.
Brett Gardner went 4-for-5 with an RBI as New York took a third win in four games against the Mariners.
Seattle starter Felix Hernandez held the Yanks to one run on five hits in a seven-inning performance.
Brendan Ryan supplied the Mariners' only RBI.
Swim Week Miami Beach 2014 [PHOTOS]