Shorn of key players in attacking areas, Spurs had to overcome a 1-0 first-leg deficit but slumped to a 3-0 defeat in east Germany and crashed out.
Marcel Sabitzer scored twice in the opening 21 minutes, Hugo Lloris culpable for both goals, to put the Premier League club on the brink of elimination and they were unable to reproduce their remarkable comeback from the same situation at Ajax in last season’s semi-final.
Emil Forsberg completed the rout late on with his first touch as Spurs bowed out with a whimper.
Defeat, on a back of a disappointing exit to Norwich in the FA Cup last week, leaves Jose Mourinho with no prospect of silverware this season.
He now has nine games left in the Premier League to try and ensure Tottenham are back in the Champions League next season, but even that is looking unlikely given their position in the Premier League table.
But the way they were unable to repel Leipzig over the two legs highlights how weak they are in defence, with goalkeeper Lloris also letting them down.
The Bundesliga side, though, managed by 32-year-old manager Julian Nagelsmann, nicknamed ‘Baby Mourinho’, moved into the Champions League quarter-finals for the first time in their history.
Mourinho had been bullish about his side’s chances of progressing before kick-off, even if he knew it would take “something special”.
Spurs have a history of Champions League revivals, most notably last season’s unforgettable comeback in Amsterdam, but, even with Mourinho’s optimism, there was a sense that it would take something similar to prevail here.
But any hope Spurs had of a fast start was dashed inside the opening 20 minutes as they collapsed and found themselves two goals down.
Leipzig cut their defence open at will, although Lloris was most at fault.
In the 10th minute, the France captain, culpable for a goal at Burnley on Saturday, failed to keep out Sabitzer’s scuffed shot from 20 yards, despite getting a decent hand to it.
Timo Werner had a goal rightly chalked out for offside, but Leipzig soon had their second and again Lloris was to blame.
Serge Aurier did his keeper no favours, misjudging a header and then allowing Angelino to get a cross in too easily, but Lloris still should have saved Sabitzer’s header, having again got a hand to it.
Spurs had a mountain to climb and the absence of key men in attack made their task even harder.
The influential Giovani Lo Celso started to get into the game and had the visitors’ first effort on goal as his low curling effort forced Peter Gulacsi into a stretching save.
Spurs had plenty of the ball in the second half, with the hosts happy to sit off and hit them on the counter-attack, a ploy which saw Patrik Schick and Werner blaze off target.
Dele Alli had Tottenham’s only chance of the second half as he shot straight at Gulacsi and Spurs were punished in the final five minutes.
Forsberg had only just come on when he was on hand to rifle home from Angelino’s pass and complete Tottenham’s misery.