Spurs produced one of the best displays of the Postecoglou era at Deutsche Bank Park on Thursday to earn a narrow win and progress 2-1 on aggregate to ease the pressure on the Australian.
The quarter-final second-leg tie had been billed as make-or-break for Postecoglou, but Tottenham delivered thanks to Dominic Solanke's 43rd-minute penalty before Vicario made a key save from Fares Chaibi after 75 minutes to help set up a meeting with Bodo/Glimt next month.
With Spurs down in 15th in the Premier League, the Europa League has long been viewed as the last chance saloon for the club to salvage this campaign, but a masterful away display in Germany has increased belief a 17-year trophy drought could still end this season.
"An amazing night for us, an amazing achievement, an amazing result," Vicario reflected.
"We go to the semi-final now, the job is not done but I saw there was a really big desire to fight to get this win and to go through this round from everyone who started, who came off the bench.
"It's a really proud night for Tottenham Hotspur.
"Everyone suffered a lot this season. It's been a very tough one but now we are still alive and we want to fight for something special.
"It's an important achievement. The job is not done but it's a big win for Tottenham Hotspur, for everyone, for the boss, for us players, for the staff, for the fans. So, we are very proud of what we showed."
A hostile atmosphere from Frankfurt's supporters greeted Tottenham, but they impressively handled being in the "lion's den" after previous difficult occasions at Galatasaray and AZ Alkmaar.
The aforementioned poor displays combined with a run of four straight away defeats contributed towards Spurs being tipped to exit the competition on Thursday and yet Vicario believes the victory showed what they are capable of with another second leg on the road in the semi-final stage on May 8.
Vicario added: "Yeah, it's just a perception, but we know what we are capable of when we are really tuned in, when we work hard as a team like we showed.
"The achievement in both legs has been very, very deserved from what we showed at our home ground and tonight with the resilience, the desire to fight. We got the job done and it's a fully deserved semi-final of a big European competition.
"We know we are going to face another tough opponent in the semi-final. We know when we arrive at this stage of the competition that every game will be tough, but we will fight for sure because tonight we showed that we have the capacity of doing that.
"For us it's very important to be through. It's a big night. Now we want to live this moment altogether because it's been a very suffering season."
