The Tannadice chairman told Dundee United's 28,000 disappointed fans who had travelled to Glasgow - some of whom have been openly critical of their team's "no show" on the day - to remember where the club was a few short years ago.
Thompson has held the reins at Tannadice for six years following the death of his father, Eddie, in 2008, and has overseen a surge in Dundee United's fortunes. But he admitted United came very close to not even seeing these recent times, given the perilous state the club was in.
"We are all gutted about [the final outcome] in Glasgow but, while there is a huge disappointment, this Scottish Cup run has been another good experience for the club," said Thompson. "Remember, this is a football club that has known some really dark times. Dundee United should actually have been placed in administration just a few short years ago.
"The first five years of my father's reign were dire. We were really struggling to keep the club's head above water. My father ploughed a million pounds a year into the club while it was losing £2.8m a year at the same time. We were haemorrhaging money.
"Hindsight is a great thing, but I look back now and think that putting Dundee United into administration the day my father took over would have been the right thing to do in the circumstances. But he wouldn't do it.
"I think all United fans need to remind ourselves of where the club has been these past 20 years. We've been way down the old SPL, we've been relegated, we've struggled badly. For years we were embroiled in the bottom six and relegation battles.
"Now look where we are - contesting for trophies, top half of the table, with some terrific young talent on the pitch. It's an exciting time, but we just have to remind ourselves of where we've been not so long ago."
Thompson revealed that United have raked in around £900,000 from their Scottish Cup run, and that he and Jackie McNamara were already hatching plans for next season. For the first time in years, he insisted, United would not need to sell anyone over the summer.
A number of United supporters have been highly critical of their team's performance at Celtic Park, though Thompson was keen to credit their opponents for their historic win.
"I would like to congratulate Geoff Brown and St Johnstone - it was a great day for them," he added. "But my aim, along with Jackie, is to make next season an even better one for Dundee United. This club is in the best place it has been in, financially, in years.
"We've got a great young team and no bank debt. We do have certain 'obligations' to the bank but these can be managed. It's nothing like as bad as it was before, when we peaked at £7m in debt.
"The good news for us is that we don't need to sell anyone over the summer. In players like Ryan Gauld, Stuart Armstrong, Andy Robertson, Gary Mackay-Steven, Nadir Ciftci - imagine how good they will be in a few years. We're gutted but we are still in a very good place."