Hitzfeld's two substitutes – Admir Mehmedi and Haris Seferovic – came on to score and lead Switzerland to a come-from-behind 2-1 win over Ecuador in their Group E opener on Sunday.
Benaglio paid tribute to former Bayern Munich and Borussia Dortmund coach Hitzfeld, who has been in charge since 2008, and the way he has developed the European nation.
"When you see how this team has developed with him coaching then you don't need many words," he said.
"What makes him special is how he speaks with his teams, something we could see during half-time against Ecuador. Then he made smart substitutions, that's because of his big experience.
"That's nothing new for us players. That's what the coach Ottmar Hitzfeld is like and, because of that, he is one of the most decorated coaches. Against Ecuador, thank God, everything went well."
Seferovic's winning goal came on the counter-attack in the 93rd minute after a last-ditch tackle from Valon Behrami at the other end.
Benaglio said the challenge from Behrami was as important as the match-winner.
"The action of Valon is nearly worth a goal," he said.
"The tackling in the own box can't be played any better and then this big effort to play the counter-attack - that was incredible.
"That shows that Valon never gives up, even if the situation he's in is difficult. Like this he rewarded himself and the team."
Meanwhile, defender Johan Djourou was pleased his team overcame early nerves to claim victory and be second to France in the group.
"We felt that the squad was maybe a little nervous," Djourou said.
"However, it shows our strength, as we were able to come back in the game and changed what was not good in order to show something completely different in second half."
