Juve went into the quarter-final second leg at Allianz Stadium on Tuesday with an away goal following a 1-1 draw in Amsterdam and Cristiano Ronaldo headed them in front in the 28th minute.
However, Donny van de Beek equalised with a cool finish before half-time and Matthijs de Ligt converted from a corner in the 67th minute to give Ajax a famous 3-2 aggregate victory over Massimiliano Allegri's side.
Juve shelled out €112million to sign five-time Champions League winner Ronaldo from Real Madrid last July in a bid to boost their chances of lifting the trophy for the first time since 1996.
Despite again failing to deliver the European success so desperately craved in Turin, Allegri – who oversaw defeats in the 2015 and 2017 finals – and president Andrea Agnelli confirmed the coach will remain at the helm next term.
Bernardeschi is confident it is the correct decision and believes Juve's wait to get their hands on the Champions League will soon be over.
"[Allegri staying is] a very important signal. I think the president will build a project to win the Champions League, just like in previous years," he told Sky Sport Italia.
"I believe the Champions League will arrive in Turin. If it's not next year, it'll be the one after that. It will come."
Allegri downplayed the size of the shock Ajax pulled off, highlighting the Eredivisie side's run to the Europa League final in 2016-17 and their last-16 victory over Champions League title-holders Madrid in March.
"First of all, Ajax deserved to go through because three out of four halves – the two halves in Amsterdam and the first half in Turin – were even. In the second half [in Turin] they did better because we lost cohesion and they created many chances," Allegri said.
"Then, maybe we conceded a goal that we shouldn't have, but overall I would say that they deserved to go through to the next round.
"Ajax have very good players, they play very good football. Two years ago they played the Europa League final. They are not newcomers. You don't score five goals against Real Madrid by chance."