A sickening incident led to Hungary striker Varga being stretchered off in front of his visibly shocked team-mates in the second half, but Csoboth's late winner kept them alive in the tournament.
The Hungarian Football Federation said shortly after Sunday's match that Varga was in a "stable condition" in a hospital in Stuttgart.
The episode brought back harrowing memories of Christian Eriksen's collapse at Euro 2020, when Denmark players shielded him from the cameras and those in the stadium after he suffered a cardiac arrest.
Csoboth dedicated the goal to Varga by holding up his stricken team-mate's shirt during his celebration.
"Of course we were fighting for him in the remaining 15-20 minutes. We wanted to win for him and we dedicate the win to him," said Hungary attacker Roland Sallai.
It was a bitter end to Scotland's bid to reach the knockout phase of a major tournament for the first time as they were caught out pressing for a goal of their own right at the death.
"It was always a one-goal game," said Scotland coach Steve Clarke. "We didn't manage to get the goal, we opened up at the end to try and get it."
Germany finished top of Group A after a 1-1 draw with Switzerland, who will go through as runners-up. Hungary must rely on other results if they are to advance as one of the best third-placed sides.
Scotland came last with just a point as their wait for a first European Championship win since 1996 goes on.
Clarke made one enforced change for Scotland as Scott McKenna replaced Kieran Tierney, who was stretchered off with a hamstring injury in the 1-1 draw with Switzerland.
English-born Callum Styles got the nod in Hungary's midfield, coming in after shaking off an injury, with Endre Botka also making his first appearance of the tournament on the right side of a back three.
John McGinn revealed the team skipped their pre-match nap as thousands of Scotland supporters gathered outside the hotel, belting out songs in anticipation of what they hoped was a historic night.
Hungary arrived at the Euros with increased expectations after a 14-match unbeaten run between November 2022 and March of this year. They also beat England twice in the last UEFA Nations League, as well as Germany away.
But back-to-back defeats had punctured confidence and left Marco Rossi's side needing to beat Scotland to keep alive their hopes of reaching the knockout rounds.
Cagey opening
Scotland goalkeeper Angus Gunn shovelled away a dipping drive from Bendeguz Bolla that bounced awkwardly in front of him, with Styles and then Willi Orban booked for cleaning out McGinn in a first half full of endeavour but limited in genuine chances.
Che Adams' dangerous clearance though led to a Hungary free-kick 25 yards out, which Dominik Szoboszlai clipped towards the far post with Orban's header skimming the top of the crossbar.
Szoboszlai, who shrugged off injury concerns after missing Hungary's final team training session, hammered over a strike from distance, and it wasn't until the 53rd minute that Scotland registered a shot.
Scotland defended doggedly as Jack Hendry blocked Sallai's attempt when he shifted in from the left before Marton Dardai headed over from a cross.
A horrifying moment left Varga prone on his back in the Scotland area as he tried to head a free-kick goalwards while Gunn came flying out, with Anthony Ralston also crashing to the ground.
It resulted in a long stoppage as medics set Varga in the recovery position before a sheet was draped around him as he was carried off on a stretcher.
Scotland saw a penalty shout controversially turned down after Stuart Armstrong went down under a challenge from Orban, before the fourth official indicated 10 minutes of stoppage time.
That sparked a desperate quest from both teams to find the winning goal.
Gunn saved well from Andras Schafer and Szoboszlai, before Csoboth hit the post. Scotland defender Grant Hanley had a chance parried away by Peter Gulacsi, with Hungary racing up the other end and Csoboth steering home Sallai's cutback to break Scottish hearts.
