Robert Rainey, known as Archie, is understood to have suffered a heart attack in one of the stands at about 18:00 local time.
Medics tried to resuscitate the 62-year-old Belfast man inside the stadium but he was later pronounced dead.
His family said he died "doing what he loved best".
In a statement, they said: "Our father has sadly passed away whilst in France supporting Northern Ireland at the Euros, surrounded by all his friends and family.
"We are obviously devastated at this loss, and thankful for everyone's thoughts and prayers."
Northern Ireland manager Michael O'Neill said he and his squad were "all shocked when we heard this sad news after the game".
"The thoughts of all the players and the backroom staff are with Robert's family and friends at this time."
The Irish Football Association (IFA) said it will request that Northern Ireland players can wear black armbands during the team's game against Germany on Tuesday as a mark of respect to Mr Rainey.
He is the second Northern Ireland fan to die in France this week.
Early on Monday morning, 24-year-old Darren Rodgers, from Ballymena in County Antrim, died after a fall from a promenade in Nice.
The incident happened just hours after Northern Ireland's opening game of the tournament against Poland.
Northern Ireland's players wore black armbands during the win over Ukraine in Lyon in memory of Mr Rodgers.
There was also a minute's applause by supporters inside the ground in the 24th minute of the game as a tribute to him.
After Mr Rainey's death, Supt Nigel Goddard, of the Police Service of Northern Ireland, said: "Our thoughts are tragically with a second family this week mourning the devastating loss of their loved one."
Patrick Nelson, the IFA's chief executive, said the man's death during the game "puts any result of a football match into perspective".
John Delaney, the chief executive of the Football Association of Ireland, said the man's death was "a sad outcome to what was a historic day for Northern Irish football".
A Northern Ireland supporter has died at the Stade de Lyon as he watched the team beat Ukraine at the Euro 2016 tournament in France.