Defending champion France lost its title and faces relegation to the UEFA Nations League second-tier after losing at home to Croatia 1-0 on Monday.
Their group leader Denmark, in the meanwhile, beat Austria 2-0 to boost its Final Four bid.
Denmark has nine points, two more than second-placed Croatia with only the group winner going through. Austria was third with four points and rock-bottom France had two after a fourth game without a win. The team in last place is relegated.
Croatia scored with a fifth-minute penalty from standout midfielder Luka Modric at the Stade de France. It is where he played so commandingly for Champions League winner Real Madrid in the final just over two weeks ago, and he belied his advancing years with a display full of astute passing and clever movement into space.
Croatia's fans chanted, "Luka, Luka,” to their inspirational captain as he gave them a thumbs up after the game.
In Copenhagen, wide forwards Jonas Older Wind and Andreas Skov Olsen got Denmark's goals in the first half.
Croatia hosts Denmark in a table-topping clash on Sept. 22.
France coach Didier Deschamps finally dropped first-choice forward Antoine Griezmann, who has not scored for the team since early January and instead selected Christopher Nkunku in attack alongside Karim Benzema and Kylian Mbappe.
It was Nkunku’s astute pass that set up Mbappe’s late equalizer in a 1-1 draw in Austria on Friday.
Perhaps to rouse the players, the band stopped playing at the start of the "La Marseillaise” and most of the 77,400 crowd at Stade de France sang it A cappella style.
The home fans fell quiet when Croatia won a penalty after Ibrahima Konate fouled Josip Brekalo for the penalty.
Mike Maignan started in goal ahead of long-standing No. 1 Hugo Lloris and guessed the right way but he could not stop Modric's powerful strike arrowing into the left corner.
France beat Croatia 4-2 in the 2018 World Cup final but the Croats were better on the night against a French side lacking intensity in midfield. Led by the craft of 36-year-old Modric, Croatia's midfield expertly cut the supply line to star forwards Benzema and Mbappe.
Benzema went close midway through the second half as France found more space and Griezmann received a loud ovation when he came on for the last 10 minutes.
Benzema had an injury-time header well saved by goalkeeper Ivica Ivusic as French hopes evaporated.
A youth ran onto the field at the final whistle and sprinted to Benzema, forcing security to intervene, but Benzema hugged the boy, removed his jersey and handed it to him.
In League B, Group 2 leader Israel twice came from behind to draw in Iceland 2-2.
In League C, Kazakhstan took control of Group 3 after beating Slovakia 2-1 at home to move onto 10 points, four ahead of the Slovaks in the second spot.
Azerbaijan beat Belarus 2-0 at home in the other game.
