Bastian Schweinsteiger was penalised for an apparent handball on the brink of half-time, allowing Antoine Griezmann to score his fifth goal of the tournament from the penalty spot.
And although Die Mannschaft controlled proceedings, they could not fashion any opportunities to find a way back into the game as time wore on.
They were eventually dealt a gutting blow when Griezmann pounced to stab his and France's second goal of the evening past Manuel Neuer on 72 minutes.
For all their efforts, the reigning world champions could not respond to the setback of Griezmann's second goal as they bowed out of the tournament.
Germany's elimination means Liverpool have no players left on international duty, with Joe Allen and Danny Ward's Wales knocked out by Portugal - who will meet France in the final at the Stade de France this weekend - on Wednesday night.
All Reds' internationals sent home from Euros after Germany's exit
At one time, the Reds had as many as 12 players at the competition - a number only matched by Juventus of any other club - but Can will now have three weeks' holidays before being expected to link up with his club teammates in pre-season.
Having hinted at a potential involvement earlier in the week, manager Joachim Löw brought the 22-year-old in for injured defensive midfielder Sami Khedira as part of three forced changes.
It was Can's first appearance at the competition, having been an unused substitute in their previous five matches, starting alongside Manchester United midfielder Schweinsteiger.
But despite showing glimpses of the immense talent he regularly showed under Jürgen Klopp at Anfield last term, Can - who was booked early on - was generally not at his peak level of performance as Germany toiled.
Can was eventually hooked on 66 minutes, Mario Götze - heavily linked with a move to Liverpool earlier this summer although rumours have recently faded - his replacement as Low looked for a way back into the game.
But the manager's changes made no tangible difference as France doubled their lead just six minutes later and held on to book their place in the final on home soil this Sunday.