The Belgian-born midfielder, 18, has put pen to paper on a new deal that ties him to the Red Devils until June 2018.
May, who was a Champions League winner with United in 1999, has watched Januzaj closely since the teenager moved to Old Trafford from Anderlecht in 2011.
"Great to see Adnan sign a five year deal fantastic player with massive potential well done David Moyes keeping him," tweeted May.
"He's gonna be one of the players who will get people excited seen him week in week out in the reserves ... Class player."
Januzaj has made three Premier League appearances for United, scoring two goals.