The trade to obtain the outfielder from the Chicago Cubs had been widely expected, but final approval and confirmation of the deal was not made until less than three hours before the game against the visiting Rays.
Soriano will bat fourth and play left field.
The Cubs, who reportedly have agreed to pay a large portion of Soriano's $18 million salary in 2014, received minor league right-handed pitcher Corey Black in return, the Yankees said in a statement.
The 37-year-old Soriano, who began his Major League Baseball career with the Yankees in 1999, hit .254 with 17 home runs and 51 runs batted in over 93 games with the Cubs this season.
The defending American League East champion Yankees (54-48) are in fourth place in the division, 6 1/2 games behind the Boston Red Sox as they have struggled offensively due to a string of injuries to their regular players.