Murat Yakin's men were a class above in the first period and deservedly took a two-goal lead into the break through Kwadwo Duah and Michel Aebischer, both of whom registered their first goals for the national team in the process.
Marco Rossi's Hungary offered little threat whatsoever during the opening period, but they belatedly lit the blue-touch paper in the second period through a Barnabas Varga header and looked the likeliest to score again in a dramatic finale.
However, a late defensive error - not the first of the day from the Hungarians - teed up Breel Embolo to restore Switzerland's two-goal advantage right at the death in a particularly poignant moment for the 27-year-old.
Embolo only managed five appearances for Monaco in the 2023-24 season due to a crushing ACL injury and subsequent muscular problem, but the striker flaunted his attacking prowess in the short time that he was on the pitch, and his cheeky lob in injury time was a fitting end to a chaotic battle.
Amid all the pre-game fanfare for Liverpool midfielder Dominik Szoboszlai, the 23-year-old earned a slice of Euros history at the RheinEnergieStadion, becoming the youngest man to ever captain his nation at the Championships.
History-making Aebischer steals Szoboszlai's thunder
However, Szoboszlai's record paled into insignificance with just 13 minutes gone, when Switzerland's fast start was rewarded, as was Yakin's bold attacking decision.
The Rossocrociati boss put his faith in Ludogorets forward Duah - who had only won one cap for the national team - from the first whistle, and the 27-year-old opened his Swiss account with a cool finish into the corner from Aebischer's exquisite through ball.
The offside flag initially halted Duah's celebrations, but a VAR review showed that his goal was good, triggering another ecstatic episode from the Bulgaria-based attacker before Yakin's men continued to dominate.
Bournemouth left-back Milos Kerkez was nearly left to lament a horror mistake in the 20th minute, overhitting a backpass to Willi Orban and instead playing Ruben Vargas through, but the shoulder of Peter Gulacsi saved Kerkez's bacon.
The Swiss did not let up in their search for a second and only had one real scare to contend with in the 40th minute, when Orban sent a free header straight at Yann Sommer from a free kick, although replays showed that he may have gone too early anyway.
Just when it appeared that Switzerland's search for a pre-half time insurance goal would not pay dividends, Aebischer - the provider of Duah's effort - etched his name into Rossocrociati folklore, curling home a stunning second from the edge of the box after a period of Swiss probing.
In doing so, the Bologna man became the first male Switzerland player to score and assist in one Euros match as Yakin's men took a merited two-goal lead into the break, after which normal service resumed.
It took two strong saves from Gulacsi to prevent Duah and Vargas from launching Switzerland out of sight in the first 10 minutes of the second half, but all of a sudden, Yakin's men went to sleep and failed to heed a serious warning.
Sommer was left flat-footed by a Varga header which flew wide in the 63rd minute, and while the Ferencvaros attacker should really have done better with that chance, he made amends just three moments later.
Latching onto an inswinging cross from Szoboszlai - who had cut a particularly vexed figure in the opening hour - Varga outmuscled Aebischer at the back post and directed a low header past the sprawling Sommer to cut the deficit in half.
Varga's contribution set up a tantalising finale, in which Rossi's men huffed and puffed with vigour - a stark contrast from their tepid first-half display - but Gulacsi had to be the hero of the hour again in the 90th minute, preventing Szoboszlai from scoring an own goal after a dangerous Switzerland corner to the near post.
In the third minute of added time, however, Hungary's valiant second-half efforts would be for nought, as a poor header back from Orban set up the injury-plagued Embolo to nonchalantly lift the ball over Gulacsi's head and into the net in front of the elated Rossocrociati faithful.
When do Hungary and Switzerland play again at Euro 2024?
Switzerland's success immediately moves them up to second place in Group A, only behind leaders Germany on goal difference, as Rossi's men sit third, above Scotland on the same metric.
Both sides are back in action on Wednesday, June 19, as the Hungarians battle hosts Germany at Stuttgart's MHPArena, while Switzerland return to the RheinEnergieStadion in Cologne to battle Scotland.