In-form forward Enner Valencia gave Ecuador a 22nd-minute lead in Sunday's first Group E encounter in Brasilia, prompting veteran coach Hitzfeld to bolster his own attack with Admir Mehmedi at half-time.
Mehmedi headed the European team level within three minutes of the restart and another substitute, Real Sociedad striker Haris Seferovic, crowned a breathless finale in dramatic fashion as he applied the finish to Ricardo Rodriguez's cross in the final minute of injury time.
Rather than taking the credit, Hitzfeld, who will call time on his decorated career at the end of the tournament, chose to praise the character his players showed during a contest that retained a frantic pace throughout.
He told the Swiss FA's official website: "I'm proud of my team. The pressure in the first World Cup game is always very large.
"I told the players to be patient and piece by piece they fought back into the game.
"We needed the three points to start off. The first step is done. And this victory is, of course, good for the confidence."
Seferovic, who endured a fallow season at club level, was pleased to show his poacher's instincts remain strong.
He said: "There is nothing better than to achieve a goal. I had, until then, not a lot of balls but when it comes to it, you have to stop to be ready."
Mehmedi echoed his coach's thoughts on the importance of Switzerland's team spirit.
"Of course I was disappointed that I was not allowed to play from the beginning, but I was ready to be used," said the Freiburg forward.
"As a striker, you have to stand where I stood at 1-1.
"Crucially, we never gave up. Morale and team spirit have been the decisive factor(s) for us."
