In an eventful first 45 minutes, Son and Harry Kane had wasted chances for the hosts, while Juventus were denied a clear penalty when Douglas Costa was brought down by Jan Vertonghen.
Tottenham appeared to be in control until Gonzalo Higuain turned in Sami Khedira's flick to give Juventus a 64th-minute lifeline - which they capitalised on ruthlessly and Paulo Dybala raced clear for the decisive goal three minutes later.
Mauricio Pochettino's side fought to revive their chances but to no avail, coming agonisingly close in the dying moments when Kane's header hit the inside of the post but bounced right on the goal-line.
For Spurs, who last reached the Champions League quarter-finals in 2010-11, it was a first defeat in 18 matches, while Juventus are unbeaten since 22 November - a run of 20 games.
They are the first English team to be eliminated from this season's Champions League. Elsewhere on Wednesday, Manchester City lost 2-1 to Basel but went through 5-2 aggregate win to join Liverpool in the quarter-finals.
Manchester United and Chelsea could make it four Premier League sides in the last eight of Europe's top club competition when they play the second legs of their ties next week.
United entertain Sevilla on Tuesday following a goalless first leg in Spain, before Chelsea play at Barcelona on Wednesday after it finished 1-1 at Stamford Bridge.
Tottenham v Juventus - as it happened
Man City into last eight despite first home loss since 2016
Heartbreak as Spurs fall short again
Pochettino has performed outstanding work in piecing together an exciting and talented Tottenham team - but familiar questions will be asked after this devastating loss.
Spurs showed maturity as well as excellence to come from two goals down to earn a draw in Turin in the first leg, putting themselves in a favourable position to finish the job and secure a place in the quarter-finals.
And yet, from a position of strength bolstered even further by Son's goal just before the interval, Spurs once again came up short.
There is no doubting the quality in this Tottenham side and they were excellent in spells at Wembley, but with two Premier League title campaigns promising much but unable to deliver and an FA Cup semi-final loss to Chelsea last season, the requirement for tangible success is intensifying.
The hosts missed chances and suffered ill luck in the closing moments when Kane's header somehow stayed out, but in the end they paid a heavy price for three minutes of slackness that undid all the fine work from the first leg.
The best sides back up quality with trophies. Spurs now need the FA Cup this season.