A Breel Embolo penalty had broken the deadlock for Switzerland early in the first half in Santa Clara, before the Swiss were made to pay by a Miro Muheim own goal.
Qatar, appearing in just their second finals after hosting the 2022 World Cup, looked rusty after their preparations were disrupted by the Middle East war, meaning their meeting with Switzerland was just their third match since December last year.
And for the majority of the match they were outclassed as Switzerland racked up 26 shots, before the unfortunate Muheim put through his own net under intense pressure from Qatar captain Boualem Khoukhi four minutes into injury time to spark wild celebrations on the Qatar bench.
“I am very proud of the team,” said Qatar’s Spanish coach Julen Lopetegui.
“I told them that even if we hadn’t scored the goal and didn’t draw I would have been proud of the mentality and discipline that they showed today.
“But fortunately we scored and that was history.”
For Lopetegui it was also a landmark moment as he coached his first World Cup match — eight years later than he thought it might have been.
The 59-year-old had been set to guide Spain at the 2018 edition in Russia, but was sacked days before the start of the tournament after it was announced he had agreed to take over Real Madrid after the World Cup.
The Swiss are seeking to progress to the knockout stages for the fourth consecutive World Cup but their inability to kill off a match they dominated will worry coach Murat Yakin.
Manuel Akanji offered Qatar the first big chance of the match in just the second minute as his defensive lapse sent Edmilson Junior through, but he could only shoot tamely at Gregor Kobel.
That let-off woke up the Swiss, who were awarded a penalty on 13 minutes as goalkeeper Mahmoud Abunada clattered into Remo Freuler, despite a suspicion of offside, and after a four-minute stoppage Embolo sent the goalkeeper the wrong way from the spot.
The rest of the opening period was one-way traffic towards the Qatar box but Edmilson nearly caught the Swiss cold just before half-time, drawing a right-footed save from Kobel at the end of a rare foray forward.
Under a blazing Californian sun the chances dried up in the second period with the most notable moments a Granit Xhaka drive from distance that whizzed narrowly over the bar and an Embolo poke that nestled in the side netting.
But ultimately Switzerland were made to pay for their profligacy as 35-year-old Khoukhi powered in at the back post and caused Muheim to inadvertently head home a 94th-minute equaliser.
Switzerland next play Bosnia on Thursday in Los Angeles, while Qatar meet co-hosts Canada in Vancouver on the same day. (AFP)