The Group B match in Copenhagen's Parken Stadium was halted after Danish midfielder Christian Eriksen's collapse near the end of the first half. After leaving the pitch on a stretcher, Eriksen, 29, was later reported to be "stabilized" and conscious in the hospital.
After UEFA resumed the game at the request of both sides, Joel Pohjanpalo, 26, scored the winning goal for Finland with a header in the 60th minute.
In Finland's first-ever EURO match, it was also their first win.
UEFA tells Eriksen get well soon
UEFA chief Aleksander Ceferin wished Denmark star Eriksen a quick recovery, saying on Twitter: "Football is a beautiful game and Christian plays it beautifully."
"Tonight's Star of the Match is Christian Eriksen. Wishing you a speedy recovery, Christian," the match organizers added.
Eriksen is a Denmark regular, having scored 36 goals in 108 international caps for his nation since his debut in 2010.
Seen as one of the most reputable players in Europe, Eriksen played for Ajax Amsterdam and Tottenham Hotspur before his move to Inter in 2020.
Eriksen helped Inter win the 2021 Italian Serie A title.