The South Americans took the lead in the first half but an Admir Mehmedi header levelled the score before his fellow substitute secured the win deep into injury time
The in-form Enner Valencia scored his fifth goal in the past five international appearances to give the South Americans the lead after 22 minutes but Admir Mehmedi levelled three minutes after coming on as a half-time substitute.
Both those goals came about as a result of poor set-peice 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 a high-tempo start to the match, 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 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.
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 backwards step following the equaliser, 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 capitalised 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 are next up for Ecuador later on the same day in Curitiba.