The forward confidently slotted into the far bottom corner midway through the opening period to seal a deserved victory for the Saints, who return to the top flight at the first time of asking.
Leeds had their moments in the capital, with Daniel James striking the underside of the bar during the closing stages, but the Yorkshire giants failed to sufficiently threaten Southampton's goal throughout the contest.
Russell Martin secures promotion with the South-coast side in his first campaign in charge, Southampton joining Leicester City and Ipswich Town in sealing places in the top flight.
Armstrong class the difference in first half
Both bosses opted for unchanged starting lineups and the continuity appeared to help their respective teams with Archie Gray sending a low effort wide for Leeds within minutes of kickoff.
Southampton had to dig deep to weather Leeds' opening storm, their backline far more resolute of late, but there was only a Will Smallbone free kick that threatened an opener at the other end in the first 20 minutes, Illan Meslier making the save.
That soon changed midway through the half as the Saints went in front. A through-ball from Will Smallbone was threaded through to Armstrong, who had expertly beaten the offside trap, and the forward retained his composure to slot an effort into the far corner from 12 yards out.
Ten minutes before the break, though, Southampton were dealt a blow when David Brooks needed to be withdrawn with a shoulder injury, Sam Edozie being introduced in his place.
For all of the pressure from Leeds that followed, without any clear-cut chance being created, Southampton almost doubled their lead before the break. A cleverly-worked free kick found the run of Armstrong, whose low shot from a similar position to his goal was parried away by Meslier before being cleared.
Southampton avoid scares to win promotion
Both teams came flying out the blocks after the restart, yet it was Leeds who should have been level. A direct run from back to front from Joe Rodon saw him combine with Wilfried Gnonto to find himself in front of goal, only for Taylor Harwood-Bellis to make a last-ditch challenge to prevent what seemed like a certain goal.
Crysencio Summerville then sent a curling effort wide of the target, but it was the trigger for a spell of sustained Leeds pressure, albeit that came to nothing.
Southampton remained a danger on the break, however, and Samuel Edozie was released down the left as the game entered the final quarter. The winger chose to bide his team before attempting to find the far top corner, only to lift his effort over the crossbar.
With 17 minutes remaining, Farke decided to withdraw Summerville - the Championship Player Of The Year - but one of Leeds' substitutes went agonisingly close to finding an equaliser.
The ball fell invitingly for Daniel James around the edge of the area, the winger taking advantage of the situation by launching a dipping, swerving volley towards the far top corner which hit the underside of the crossbar and was cleared.
During nine minutes of added-on time, James fired another effort towards goal, goalkeeper Alex McCarthy doing well to parry his strike away from danger.
That proved to be Leeds' final chance, Farke's side falling short at the second time of asking having initially lost their way during the run-in of the regular season, and they are left to contemplate another battle in the second tier in 2024-25.