The Major League Soccer team announced the signings Monday without disclosing terms.
Robles was the only Red Bull to play every minute this past season. The 29-year-old recorded 11 shutouts and a 1.21 goals against average in helping New York win its first Supporters' Shield. He was tied for first in the league in wins (17), third in saves (103) and fourth in shutouts.
Miller is the longest continuously tenured Red Bull, with the club since January 2010. He appeared in 17 games and had one assist before injuring his Achilles' tendon in late August, which sidelined him for most of the remainder of the season.