That brought up Swisher, who drilled a pitch into the second deck in right field for his 200th career round-tripper and a 4-2 Yankees lead.
The New York designated hitter later added a run-scoring single while Eric Chavez went 2-for-4 with a solo shot as the American League East-leading Yankees took the opener of a four-game set.
David Phelps (3-3) tossed five innings of two-run ball to earn the victory, allowing six hits and walking one while striking out three.
Sixteen-year veteran hurler Derek Lowe, signed by the Yankees earlier in the day, collected his first save since 2001 by finishing the game with four shutout frames.
Texas starter Ryan Dempster (1-1) was rocked for eight runs on nine hits over six full innings.
David Murphy socked a solo homer for American League West-leading Texas.