Armstrong's first-half strike at Wembley ensured Southampton erased the bitter taste of last year's relegation from the top-flight with an immediate return.
Victory in football's single richest match will give Southampton an estimated £140 million ($178 million) windfall due to the increases in matchday, broadcast and commercial revenue for clubs who reach the Premier League.
Financial experts Deloitte say Southampton could eventually bank £305 million if they avoid relegation next season.
While the cash prize is a huge bonus, it is the prestige of competing with Manchester City and Arsenal, rather than second-tier minnows like Plymouth and Oxford, that fuelled Southampton's hunger to regain their Premier League status after a year in purgatory.
Southampton, whose only major honour is the 1976 FA Cup, had spent 11 years in the Premier League prior to their relegation.
They will hope this success, celebrated by over 35,000 of their fans at Wembley, is the prelude to another lengthy spell among the elite.
In their first ever play-off final, Southampton made it three wins over Leeds this term despite finishing three points behind the Elland Road club in the regular season.
Saints boss Russell Martin has become widely admired for his purist passing principles and this was the 38-year-old's crowning glory.
Martin was managing in League One with Milton Keynes Dons just three years ago, but he has transformed Southampton in his first season in charge, to such an extent that they boasted more possession than any team in Europe's top five leagues this season.
"I'm so pleased for the board because I was not an exciting appointment for them. I finished 10th at Swansea, but we had a really clear way of doing things at MK (Dons) and here, so for them to take me on was really brave," Martin said.
"We had to do this to justify their decision. I feel really emotional about it.
"The team have been so brave. I'm really grateful and proud of them. They deserve it."
Securing promotion at Leeds boss Daniel Farke's expense was a sweet moment for Martin after his Norwich contract was terminated by the German during their time together at Carrow Road in 2018.
Defeat was a bitter blow for Farke and Leeds, who were relegated alongside Southampton last season.
Back to the promised land
American socialite Paris Hilton and movie star Will Ferrell -- a minority shareholder in the club -- had made good luck videos for Farke's side, but there was no Hollywood ending for Leeds.
On the 35th anniversary of the death of their legendary manager Don Revie, Leeds were well below their best in the club's fourth play-off final loss.
After losing their grip on an automatic promotion place that went instead to Ipswich, the three-time English champions remain without a win at Wembley since 1992.
Leeds made a fast start but Southampton weathered the storm and snatched the lead in the 24th minute.
Armstrong had scored twice in the play-off semi-final win over West Brom and the striker showcased his predatory instincts again with his 24th goal this season.
Cleverly eluding the Leeds offside trap with a well-timed run, Armstrong met Will Smallbone's pass with a superb strike into the far corner from 12 yards.
Leeds were rocked and Armstrong nearly struck again from another instinctive shot from Smallbone's quick free-kick, but this time Illan Meslier was able to parry his close-range effort.
Early in the second half, Leeds defender Joe Rodon embarked on a rampaging run that looked certain to end in a goal until Taylor Harwood-Bellis threw himself into a last-ditch block.
Leeds pushed hard for an equaliser, but Daniel James' blast cannoned back off the bar before the winger was denied by Alex McCarthy's stoppage-time save as Southampton celebrated their return to the promised land.
