A year after hitting .273 in a career-high 91 games, the versatile Forsythe hit .214 while starting games at five different positions with the Padres in 2013.
The 27-year-old played in 75 games overall, and was one of six major leaguers to play four different positions while appearing in at least 10 games. The others were San Diego's Alexi Amarista, Philadelphia's Freddy Galvis, Detroit's Don Kelly, Justin Turner of the New York Mets, and Skip Schumaker of the Los Angeles Dodgers.
Boxberger, 25, split time between San Diego and Triple-A Tucson in 2013, appearing in 18 games and compiling a 2.86 ERA with 24 strikeouts in 22 innings over five stints with the Padres.
Torres went 4-2 with a 1.71 ERA in 39 appearances with the Rays. The 26-year-old left-hander ranked third among AL relievers with a .159 opponents' batting average.
Hahn, 24, went 2-1 with a 2.15 ERA in 19 starts at Class-A Charlotte last season.
The Rays also received minor league pitchers Matt Andriese and Matt Lollis, as well as infielder Maxx Tissenbaum in the deal completed Wednesday.