The Swiss held Argentina for 90 minutes, and for much of extra-time, before Angel Di Maria’s late winner finally broke their resistance.
It took a moment's brilliance from Lionel Messi to prise open the Nati defence and, ultimately, settle the contest as he played in Di Maria for the winner, but Hitzfeld insists his side can leave the World Cup with their heads held high despite the Barcelona man's intervention.
"We know Messi, in one second, can decide a match," Hitzfeld told reporters. "He has that quality."
"Pride prevails today," he continued in his post-game press conference. "The team showed a passionate performance and we were tactically very mature.
"We managed to stay calm. We saw everything that can happen in the life of the coach: the shock of the goal against us and our incredible reaction to go for it all. This is something that will make us proud."
Hitzfeld, who revealed at the end of last year that he would be retiring after the World Cup, ends his career with his "head held high" but "with a heart full of emotions".
The veteran coach endured the death of his brother, 82-year-old Winfried, on the eve of the last-16 match, a contest that Hitzfeld had described as "the highlight" of his illustrious career.
Since hanging up his boots in 1983, the former West German international has taken charge of clubs in Switzerland as well as Borussia Dormund and Bayern Munich in the Bundesliga.
He won seven German titles over 13 years, as well as the Champions League with both clubs, in 1997 with Dortmund and in 2001 with Bayern. He took the Switzerland national team hotseat in 2008 and guided the team to the 2010 World Cup two years later.
Former Lazio boss Vladimir Petkovic will now take charge of the team as they now turn their attentions towards the qualification campaig for Euro 2016.