Germany finished as Group A winners after Niclas Fullkrug netted an injury-time equaliser to deny Switzerland and earn a 1-1 draw.
The hosts had cruised through their opening two games, thrashing Scotland 5-1 and easing past Hungary 2-0, but were wasteful against the resilient Swiss, who went unbeaten in the group stage and secured second spot.
Germany will face the runner-up of Group C, which could be England, in the last 16 but as things stand it would be Denmark while Switzerland take on second place in Group B - Italy, Croatia or Albania.
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Robert Andrich thought he had fired Germany into a 17th-minute lead with a long-range shot that bounced over the hands of Yann Sommer, who was relieved to see the goal ruled out for Jamal Musiala's foul on Michel Aebischer in the build-up following a VAR check.
Germany had controlled the match but were sucker-punched by Dan Ndoye's opener just before the half-hour mark. Remo Freuler capitalised on right-back Joshua Kimmich being caught out of position and crossed low for Ndoye to stab home.
Switzerland could have been two ahead just a minute later when Ndoye sped past Antonio Rudiger to go through on goal before shooting across Manuel Neuer and inches wide of the right post.
Manchester City defender Manuel Akanji was outstanding in leading the visitors' defence, his block denying Joshua Kimmich a certain goal in the second half with Switzerland set to top the group.
The offside flag denied them a second after substitute Ruben Vargas had found the net, setting up a frantic final 10 minutes that saw Kai Havertz clip the top of the crossbar with a header, Swiss midfielder Granit Xhaka denied by Manuel Neuer before substitute Fullkrug delivered the telling touch by nodding in a crucial 92nd-minute equaliser.
What they said...
Germany manager Julian Nagelsmann:
"It's always important to have players on the bench who can change games. We have players who can come in off the bench who are different. We need every player. They can be important for us in every situation.
"[Fullkrug] gives a good argument for both jobs - coming from the starting XI and off the bench. It's good luck and bad luck for him that he is so good in that role."
Switzerland manager Murat Yakin:
"I am very proud of the performance but it's rather sad for the team that we could not take the victory home. But there was a lot of pressure from the Germans so we can say we take the draw. I am happy we didn't take a defeat.
"It wasn't enough for first place [in the group] but I think the next opponent also has to watch out for us. We can be very happy."
Fullkrug delivers potentially defining moment for Germany
Sky Sports' David Richardson:
Germany were given a timely reminder that Euro 2024 glory won't come easy - but Niclas Fullkrug delivered what might be a defining moment in their tournament.
Having cruised to victories over Scotland and Hungary, the hosts struggled to break down a resilient Switzerland, experts in reaching the knockout stages with this their sixth consecutive group stage success.
Manuel Akanji and Granit Xhaka led by example in soaking up Germany's attacks while Breel Embolo and Dan Ndoye were a constant threat on the counter-attack. It was so nearly the perfect performance to secure top spot in Group A.
That was until substitute Fullkrug rose in injury-time to plant a textbook header into the top left corner and secure Germany what should be an easier path in the knockout stages.
Finishing second would have likely landed them Italy in the last 16. Instead, it should be Denmark, Slovenia or Serbia unless England slip up on Tuesday in Group C.