Germany's Jonas Hector placed in the winning spot-kick as the reigning world champions edged past Italy 6-5 in the penalty shootout after a 1-1 draw after extra time on Saturday night.
Simone Zaza, Graziano Pelle, Leonardo Bonucci and Matteo Darmian failed to convert for Antonio Conte's men while for Germany Thomas Mueller, Mesut Ozil and Bastian Schweinsteiger missed.
Hector's last shot crept underneath the diving left hand of Buffon after which the veteran could not hold back the tears.
"Everyone has their own reasons to cry," Buffon was quoted as saying by Rai Sport. "I'm sorry because we deserved it for the group that we are, for what we have made people believe.
"It's rare for a team to miss three penalties and then that's still not enough to win."
Earlier in the game, Ozil had given Germany the lead which was cancelled out by Bonucci in the 78th minute of the game through a penalty conversion.
Buffon further added he was proud of his team which was quite brilliant in the tournament edging out the likes of Belgium and Spain.
"Regrets can arrive when you feel you didn't give everything but that's not the case. We were a step away from a great feat," said the 38-year-old 2006 World Cup winner.
"I'm sad for the fans who were always with us. We were compact and hard to beat but destiny wasn't with us this time."