Sam Allardyce's men came into the weekend needing a win to stand any chance of a late twist to what has been a forgettable campaign, with favours needed elsewhere too even if they did pick up maximum points.
Victory was also crucial for Tottenham, who were looking to leapfrog Brentford and Aston Villa into the last Conference League qualifying spot and end their underwhelming campaign on a high.
Leeds have spent the last two seasons in the English top flight after Marcelo Bielsa guided the West Yorkshire outing to promotion in 2020, but will now be playing in the Championship again next season.
The hosts were found wanting inside the opening 92 seconds into the game as Son Heung-min set up Harry Kane, who smashed home his 29th league goal of the season and handed the visitors a dream start.
In doing so, Kane equalled Andy Cole's Premier League record of scoring in 26 separate matches, a record set over a 42-game season in the 1993-94 campaign.
Leeds United came roaring back and almost grabbed an immediate equaliser, but Robin Koch failed to steer Rodrigo's pin-point cross on target before Pascal Struijk saw his shot blocked by Yves Bissouma.
Ryan Mason's side came close to a second against the run of play as Bissouma's wayward shot was picked up by Kane, but a poor first touch by the 29-year-old allowed Luke Ayling to shield the ball for Joel Robles to make the save.
Leeds regained control and kept knocking on the door, with Adam Forshaw's venomous half-volley looking goalbound until Davinson Sanchez made a vital block to turn the ball behind for a corner.
The hosts' poor showing in front of goal was summed up in stoppage time as Rodrigo's close-range header flew wide of the target before Rasmus Kristensen failed to keep his first-time shot over the top.
Despite going into the break ahead at Elland Road, Tottenham's 13-year stint in European competitions was on the cusp of coming to an end, with seventh-placed Aston Villa 2-1 up at home to Brighton & Hove Albion in their corresponding final-day showdown.
The second half kicked off in an identical fashion as the first for Leeds United, who found themselves two goals down as a final nail was hammered into their three-year Premier League spell.
Having conceded just 92 seconds into the first half, Tottenham's second of the game came 79 seconds after the restart as Pedro Porro smashed home his first goal since his January switch following a brilliant move from Kane.
Allardyce threw the kitchen sink in search of a way back into the game, with Brenden Aaronson, Georginio Rutter and Junior Firpo joining the action.
Those changes initially paid off for Leeds, whose period of sustained pressure was rewarded in the 67th minute when Rutter did well to hold off Clement Lenglet and find Jack Harrison, who drilled a low strike into the bottom corner.
However, Kane restored the visitors' two-goal advantage with a neat finish to mark his fifth goal in six games against Leeds and take his league tally to 30 for the season.
Substitute Lucas Moura then put the icing on the cake with a brilliant strike in the final minute of additional time, scoring on his last game for the London outfit following a five-year stint.
Despite a resounding performance, what has been a largely disappointing campaign for Tottenham ended in even more disappointment as they failed to secure European football after Aston Villa held on to see out a 2-1 victory over Brighton at Villa Park.