[6] On December 5, 2013, Díaz signed a minor league contract with the Boston Red Sox organization.

He split 2014 between the High–A Salem Red Sox and Double–A Portland Sea Dogs, posting a combined 1.80 ERA with 56 strikeouts and 7 saves across 50 innings.

[7] Díaz split the 2015 season between Portland and the Triple–A Pawtucket Red Sox, compiling a 1.73 ERA with 66 strikeouts and 6 saves over 37 appearances out of the bullpen.

[13] On May 5, he was promoted to the major leagues to replace Michael Feliz, who went on Family Medical Emergency leave.

[16] On March 29, 2018, the Angels placed Díaz on the restricted list after he failed to report for spring training due to visa issues.