Late substitute Kevin Csoboth scored his first international goal in the 10th minute of stoppage time as Hungary secured third spot in Group A, behind first-placed Germany and Switzerland.
Hungary now face an anxious wait to see if their three points will be enough to reach the round of 16 at Euro 2024 as one of the four best third-placed teams.
Germany topped the group after a late goal saw the hosts rescue a 1-1 draw against Switzerland.
Scotland were eliminated after finishing last in the group, with just one point from a draw against Switzerland.
"Tonight and for a long time we have to get over this,'' Scotland captain Andy Robertson told the BBC.
"It's a tough one. It's devastating. All the lads are absolutely gutted. I will say thank you to all the supporters; sorry for letting you all down."
If Hungary progress, it appeared unlikely that forward Barnabas Varga will take part after a serious-looking injury.
Medical staff treated Hungary's Barnabas Varga beside the pitch after his injury against Scotland. (AP PHOTO)
Varga was in stable condition in hospital after colliding with Scotland goalkeeper Angus Gunn.
Despite both teams needing the win in Stuttgart, chances were few and far between.
Scotland were dominating possession but not doing much with it. The Scots ended the half with 64 per cent possession but no attempts on goal. Hungary had five.
Steve Clarke's side finally had a shot in the 53rd minute, but Che Adams curled his effort over the bar.
Csoboth was brought on in the 86th minute and almost had an immediate impact, hitting the far post.
Both goalkeepers had to make saves in a chaotic 10 minutes of stoppage time before Csoboth snatched the win.
In Frankfurt, they all jumped for the ball in a frantic late push for the goal that would save Germany from their first loss at the tournament, against an impressive Switzerland.
Defender Nico Schlotterbeck leaped up, next to imposing centre-back Antonio Rudiger, who reached highest of all. In front of them both was substitute Niclas Fullkrug.
Germany's Niclas Fullkrug scored the equaliser against Switzerland in Frankfurt. (AP PHOTO)
The German trio was circled by three Swiss opponents trying to protect a 1-0 lead two minutes into stoppage time. Six pairs of feet off the ground in unison, straining to be the one to head the ball.
It was Fullkrug who won the contest, guiding the ball into the top corner of the net. The match ended 1-1.
Germany had already advanced to the round of 16 before kickoff on Sunday (local time), but a draw felt like a win in the circumstances.
Switzerland had impressed while protecting a 28th-minute goal from Dan Ndoye, and were denied a second goal late by a marginal offside ruling.
The point for each team ensured Germany finished top and Switzerland second in Group A.
Germany now go to Dortmund for a round-of-16 game on Saturday against the runners-up in Group C, currently led by England.
Switzerland head to the Olympic Stadium in Berlin to open the round of 16 on Saturday against the runners-up in Group B, probably Italy or Croatia.
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