Suzuki preceded the blast with a run-scoring single for the Yankees, who clinched the three-game set with the victory.
Phelps (4-3) gave up just one hit and worked around four walks, striking out seven on his way to a third triumph in four starts.
Reliever Joba Chamberlain surrendered a three-run homer to Drew Stubbs in the seventh, but David Robertson tossed a scoreless eighth and Mariano Rivera nailed down the win with his American League-best 21st save.
Tampa Bay's Scott Kazmir (3-3) absorbed the loss, allowing four runs on seven hits over six full frames.
Swim Week Miami Beach 2014 [PHOTOS]