Robbie Neilson's side face Rangers in Saturday's final at Hampden Park aiming to end a 10-year wait for the trophy.
It is a world away from last season when they crashed out to Highland League champions Brora Rangers 2-1 in one of the biggest shocks in the competition's history.
Savage has helped spearhead the comeback at Tynecastle, having joined in December 2020, and knows the importance of previous failures.
"It was embarrassing, getting knocked out of the Scottish Cup, and then three days later we played Queen of the South and got beaten at Tynecastle and there were protests," he said.
"What was great for me was when I spoke to Robbie about what he was saying 'Joe, this is what happens at a big club. You have to win games.' I'm here to win games.
"Those defeats allowed us to reset and strip it back and say, 'what do we need to do?' It became a blessing in a weird way.
"We didn't want that result in the Scottish Cup to happen, it was very embarrassing for us, but it allowed us to hit the reset button and Robbie changed the tactics, style of play and flipped it on its head.
"Fast forward a year. We're in the Scottish Cup final, we've finished third in the league and we're in Europe for the first time in six years.
"I'm so proud I've been part of this journey."
Savage came from Preston, where he was head of recruitment, after Hearts had been demoted to the Scottish Championship when the Premiership was halted with eight games left, with the club bottom, due to the coronavirus pandemic.
They returned at the first attempt, opening this season with an 11-game unbeaten run, with ambitions of upsetting the Old Firm dominance.
"Everyone thought it was chaotic behind the scenes before I arrived but that's not what I found. I found good staff, a high level of skills, and people who were really keen to do well," said Savage, who turns 38 on Sunday.
"Ann and Andrew (McKinlay, chief executive) had brought Robbie and his staff back to the club, which is a great appointment. I was delighted with what I inherited.
"This season we always had the ambition of being competitive. We felt we could have a team which could compete for top six and certainly try to get around the top four.
"I wouldn't say we have exceeded our own expectations, we were always confident if we played the way we wanted we'd be a good side.
"The club is on the up, not just the men's first team, the academy and women's team are thriving. It's been a great year for us, it's just hoping we can win the Scottish Cup to really cement that."
They face a Rangers side hurting from Wednesday's Europa League final penalty defeat to Eintracht Frankfurt.
Yet their run to the final has helped raise the profile of Scottish football within Europe - something Savage believes can only be good for the country.
"It helps the nation as a whole to say we have good football teams, we do play good football and can win things," he said. "That knock-on can help us get better players to come here because they see us playing against Celtic and Rangers.
"Saturday is difficult. They know the Scottish Cup is the last chance to win something. We just need to turn up and perform to our best. Even if we do, we are coming up against a really good side.
"We've not won it for 10 years. We had our player of the year dance a couple of weeks ago and we had the 2012 team inducted into the Hall of Fame. It was great for our current players to see that, the standing ovation, the feeling of the fans.
"We want to create our own history. We want to show we have created a good team and go and win these competitions.
"We've had a good season and, this is what Robbie has been saying to everyone, can we make it a great season?"