The South Americans took the lead in the first half, but an Admir Mehmedi header leveled the score before Haris Seferovic secured the win deep into stoppage time.
A pair of inspired substitutions from coach Ottmar Hitzfeld earned Switzerland a 2-1 comeback World Cup win over Ecuador in Group E.
The in-form Enner Valencia scored his fifth goal in his past five international appearances to give the South American team a 22nd-minute lead, but Admir Mehmedi leveled three minutes after coming on as a halftime substitute.
Both those goals came about as a result of poor set piece marking, but Switzerland took the points in dramatic fashion courtesy of a clinical counter-attack deep into injury time.
Mehmedi’s fellow replacement Haris Seferovic, who last netted for his club Real Sociedad in November, was on hand to convert Ricardo Rodriguez's low cross to spark wild celebrations from the Swiss.
Ecuador made high-tempo start, launching attack after attack during the opening minutes but the Switzerland defence stood firm.
Their lone striker, Josip Drmic, had a sight of goal in the 12th minute, only for Jorge Guagua to make a timely challenge as the Europeans found their feet.
Granit Xhaka and Xherdan Shaqiri tried their luck from outside the box - the latter's strike grasped at the second attempt by goalkeeper Alexander Dominguez, who then spilt a 19th-minute shot from full-back Rodriguez behind.
And so it was slightly against the run of play when Valencia headed home Walter Ayovi's pinpoint delivery as Switzerland's customary set-piece solidity deserted them.
Valencia has played a more forward role in the Ecuador team since the death of Christian Benitez last July and, as part of their celebrations, the Ecuador players pointed to the sky, apparently in tribute to their late team-mate.
Dominguez showed more assured handling when Valon Behrami headed Shaqiri's corner goalwards and the Ecuador goalkeeper then got a vital touch to send Gokhan Inler's low 36th-minute drive around the post.
Ecuador looked to extend its lead two minutes before the break but Johan Djourou made last-ditch block to deny Jefferson Montero after Felipe Caicedo had played the winger in.
Mehmedi was introduced at the resumption and Hitzfeld's decision to bolster his attack paid almost instant dividends.
Just as Switzerland's marking was found wanting for the game's opening goal, Ecuador allowed the Freiburg forward far too much room to head Rodriguez's corner into the net.
Both teams refused to take a backward step following the equalizer, Valencia remaining heavily involved as he weaved inside to ripple to top of the netting with an hour played.
Ecuador's pace on the counter-attack became an increasing threat and Diego Benaglio made a fine save after his defenders backed off the rampaging Montero, while Drmic scrambled the ball home from an offside position at the other end with 20 minutes to go.
Shaqiri drilled into the side netting with team-mates well placed inside him, while Valencia almost capitalized on a rush of blood from Benaglio, but the goalkeeper was let off the hook.
Ecuador sub Michael Arroyo erred in delaying his shot in stoppage time, allowing Switzerland to break and after Behrami had made a superb tackle to set them away, Seferovic made no such mistake when he fired home the winner with virtually the last kick.
Switzerland must tackle France in Salvador on Friday, while Honduras is next up for Ecuador later the same day in Curitiba.