Madrid got the better of city-rivals Atletico Madrid at the Stadio Giuseppe Meazza in Milan in the final of the prestigious European competition.
Los Blancos took the lead after just 15 minutes when Spain international defender Sergio Ramos scored from close range.
Yannick Carrasco equalised for Atletico in the 79th minute, as the final went into extra time and eventually penalties.
Madrid won the penalty shootout 5-3 to become the champions of Europe for the 11th time in history.
Ozil, who was on the books of Madrid from 2010 until 2013, has congratulated his former club on their Champions League triumph.
Ozil had a good 2015-16 season for Arsenal on a personal level. The Germany international attacking midfielder scored six goals and created 144 chances in 35 Premier League appearances for the Gunners.
In eight Champions League matches this past campaign, Ozil scored two goals and created 16 chances for the North London outfit.