Heung Min-Son had put Tottenham 3-2 up on aggregate in the first half but two second half goals from the Italian giants proved their undoing.
Pochettino has turned Spurs into one of the most exciting young teams in Europe and many had tipped the North Londoners to progress after their excellent 2-2 comeback draw in Turin.
Pundits and supporters were in agreement that Spurs were the better side over the two legs, although that is of little consolation to Tottenham's players and fans.
While the game hinged on a three minute spell during which Spurs conceded the two goals, they had chances late on to rescue the tie.
The best of those was when Harry Kane's header bounced off the post and along the line.
With Gigi Buffon stranded, the ball was begging to be tapped in but Erik Lamela was beaten to it by Andrea Barzagli.
"Seeing Lamela there it does beg the question if (Dele) Alli was in that situation he would have got there," Lineker said on BT Sport.
"With his goalscoring instinct he would anticipate it and be on top of it perhaps more so than Lamela was."
Alli was withdrawn for Fernando Llorente late on much to the midfielder's disappointment as he left the field shaking his head.