France fought back twice through Kylian Mbappe (23), who erased Argentina's 2-0 lead in 97 seconds in normal time before scoring a third in extra time to level it at 3-3.
But the South Americans won 4-2 on penalties.
"We are very disappointed of course, we gave everything. We had to face a lot of obstacles during the competition, we gave up nothing," Varane said.
"For an hour we were not in the game. We could have won too. I'm very proud of this group and of being French. We keep our heads high. We came back in this game when it was complicated.
"We pushed and we believed until the end. We almost turned the game around, which was not going well... There was mental strength in this group, a lot of heart. That allowed us to come back in this game. We're disappointed but also proud."
Argentina were the better team in the first half and stormed into a 2-0 halftime lead before Mbappe brought them level with goals in the 80th and 81st minute of the contest which France skipper Lloris described as a "boxing match".
"The Argentines got off to a great start, being aggressive, using all facets of the game. We were a bit reactionary all night," the goalkeeper said.
France had also come into the final after several players missed training sessions due to illness over the past week, but Lloris refused to use that as an excuse.
"We're not going to make excuses, we gave everything," he added. "You have to congratulate the guys because we didn't give up until the end."
