Martinez, the former Mariners great who is now the team's hitting coach, said in a statement that Cruz "is truly deserving of the honor."
"I see first-hand how his presence both in the clubhouse and in our lineup provide stability that makes his teammates around him better. While he goes about his work in a quiet way, we are glad that he is receiving the national recognition that he deserves," Martinez said.
Cruz, 37, finished last season with a .288 batting average, 39 home runs and an American League-leading 119 RBIs in 155 games.
The designated hitter award was renamed in 2004 in honor of Martinez, who won two AL batting titles and has the highest career batting average, on-base percentage and OPS among players with at least 3,000 plate appearances as a DH.
The honor has been awarded since 1973. Former Boston Red Sox slugger David Ortiz won the award in 2016.