Spurs' slender victory ends a 17-year wait for another piece of silverware after their 2008 EFL Cup win.
It also confirms the club's participation in the Champions League next season, as well as their place in the Super Cup with either Inter or PSG.
The only goal of the match was scored just before half-time, when Brennan Johnson guided the ball to the back of United's net.
Spurs' triumph is notable as, next season, they will be the first Premier League club to play in the Champions League after finishing outside the top ten.
Rarely has the final of Europe’s secondary competition had so much at stake, and neither defence looked particularly assured in the opening exchanges.
Tottenham worked the first clear-cut chance when Johnson made a surging run down the right and picked out Pape Matar Sarr, whose effort was blocked by Harry Maguire.
Down the other end, Maguire teed up Amad Diallo to fire a shot right across the face of goal after Spurs failed to clear their lines from a United corner.
However, Ange Postecoglou’s side broke the deadlock in the 42nd minute when a whipped cross from Sarr caused havoc in the area - with Johnson getting to the ball at the near post before Luke Shaw inadvertently applied the final touch.
It won’t win any prizes for its aesthetic qualities, but it came at a crucial time towards the end of a first-half slugfest in the Basque Country.
Johnson's breakthrough was the 31st time United had conceded the opening goal in a match this season, the most of any Premier League club across all competitions.
It was the latest damning statistic for Ruben Amorim’s men, but they looked to change the narrative when Rasmus Højlund headed over from Amad’s cross.
Tottenham’s defending hardly inspired much confidence, with Guglielmo Vicario just about managing to thwart Leny Yoro from Bruno Fernandes’ set piece.
The north Londoners were holding onto their slender lead by the skin of their teeth when Vicario failed to collect a routine free-kick from Fernandes, but he was bailed out by Micky van de Ven’s stunning acrobatic clearance of Højlund’s header.
United continued to look the more likely side to strike next, and Fernandes should have done much better when his diving header went wide from Noussair Mazraoui’s cross.
Amorim introduced Alejandro Garnacho as part of a double-change with Joshua Zirkzee, with the former almost making an immediate impact after forcing Vicario into a fine one-handed save.
Ultimately, it was to be Tottenham’s night - as Postecoglou’s players earned a piece of silverware that more gifted Spurs teams during Mauricio Pochettino’s tenure failed to deliver.
It was a fine way for the Australian to mark his 100th match in charge, as Spurs achieved their 150th victory in European competition.
Meanwhile, United’s unbeaten Europa League run ends in their 15th and final game of the competition as the inquest into the club’s disastrous campaign begins.
Flashscore Man of the Match: Micky van de Ven (Tottenham Hotspur)
