The deal was announced Tuesday, one day after former Diamondbacks outfielder J.D. Martinez agreed to a $110 million, five-year contract with Boston, pending a physical.
Drury gives the Yankees a new option at third base or second, where New York was projected to start a pair of rookies.
Arizona pitching prospect Anthony Banda goes to the Rays, who also get minor league second baseman Nick Solak from the Yankees. Minor league right-hander Taylor Widener goes from New York to the Diamondbacks.
Drury played mostly second base for Arizona but came through the minors as a third baseman and could fill that spot for the Yankees, who have top prospect Gleyber Torres penciled in at second. Another prospect, Miguel Andujar, was a leading candidate at third.
Souza helps fill the outfield power void left in the Diamondbacks' lineup by Martinez's departure.
Drury hit .267 with 13 home runs and 63 RBIs in 135 games with Arizona last season. Souza batted .239 with a career-high 30 home runs and 78 RBIs last year.
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