The supporter fell from the second tier of the main stand into the lower seating area in the final minutes of the match, ESPN reported.
"There was a medical emergency at the Munich arena, and despite the best efforts of the medical team, the person sadly passed away at 00:06. Our thoughts are with their loved ones at this difficult time," a UEFA spokesperson said in a statement.
Portugal defender Nuno Mendes said: "I would like to express my condolences to the family of the fan that died. It casts a pall on our victory today."