The deal comes after a flurry of moves involving Oakland in November and December.
Some of those transactions appeared to signal a shift toward rebuilding for Oakland.
Over nine seasons in Tampa Bay, Zobrist, 33, batted .264 with 114 home runs and 511 RBI while earning two All-Star nods (2009, 2013). Zobrist can play several positions, including in the outfield and second base.
Over 137 games with Tampa Bay last season, Escobar batted .258 with seven home runs and 39 RBI. The 32-year-old has also had stints with the Atlanta Braves and Toronto Blue Jays.
For Tampa Bay, Robertson was considered one of Oakland's top prospects, while Powell batted .343 with three home runs and 28 RBI in the minor leagues last season.
In 99 games with the A's last season, Jaso batted .264 with nine home runs and 40 RBI. The 31-year-old spent the 2010 and 2011 seasons with the Rays.
Oakland went 88-74 last season before falling to the Kansas City Royals in the American League Wild Card game, while Tampa Bay finished 77-85.