The in-form Enner Valencia scored his fifth goal in the past five international appearances to give the South American team a 22nd-minute lead 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-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.
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.
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.
