Myers, who did not make his major league debut until June 18, was announced Tuesday as winner of the AL rookie of the year award.
Myers got 23 of 28 first-place votes.
Angels outfielder Mike Trout was a unanimous winner last season.
Myers, 22, became the third Tampa Bay player to win the award in six years, following pitcher Jeremy Hellickson in 2011 and third baseman Evan Longoria in 2008. The Rays acquired Myers last winter, in a trade that sent veteran pitchers James Shields and Wade Davis to the Kansas City Royals.
Myers batted .293 in 88 games, with 13 home runs. He led AL rookies with 53 runs batted in. He batted second, third and fourth in the final weeks of the season, as the Rays qualified for an AL wild-card spot. He also went one for 16 in the Rays’ division series loss to the Red Sox.
Iglesias, 23, batted .303 with three home runs in 109 games. He played shortstop and third base for the Red Sox before he was traded to Detroit, where he replaced the suspended Jhonny Peralta at shortstop.
Archer, 25, went 9-7 with a 3.22 earned-run average in 23 starts for the Rays.
Shuck, 26, batted .293 for the Angels, with two home runs in 129 games.