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View image in fullscreenDonnarumma denies Liverpool and Núñez to send PSG through on penalties
Luis Enrique exploded across the Anfield pitch when Désiré Doué struck the winning penalty and was still leaping on Paris Saint-Germain players and officials when they headed down the tunnel five minutes later.
Mohamed Salah was the only Liverpool player to convert in the shootout but his performance over 120 minutes was wasteful in the extreme.
Alexis Mac Allister presented Salah with a glorious chance to open the scoring and double Liverpool’s aggregate lead with four minutes gone.
The three-time winners Inter last lifted the trophy in 2010, beating Bayern in the final in a treble-winning season, and Inzaghi's side are still on course to repeat that feat, with a Coppa Italia semi-final also to come in April.
Feyenoord came into the game with an uphill task after losing 2-0 at home last week, and by the eighth minute they were left with a mountain to climb following Thuram's superb solo goal. Thuram, who also opened the scoring at De Kuip in the first leg, collected the ball on the left wing before cutting inside with some clever footwork before curling a shot into the far top corner.
Mehdi Taremi sent a shot straight at the keeper as Inter looked to put the tie even further out of reach but Feyenoord pulled one back three minutes before the break from the penalty spot.
Calhanoglu was penalised for a foul on Jakub Moder who picked himself up and sent his spot kick into the bottom corner to beat Yann Sommer, just the second goal Inter have conceded in this season's competition, and the first at San Siro.
Any hope of a Feyenoord comeback all but ended when Calhanoglu made up for his mistake by converting his own penalty six minutes into the second half after Thomas Beelen fouled Taremi.
With Feyenoord taking risks in an effort to get something from the game, Inter were always dangerous on the counter and looked like they had another penalty but after a VAR check, Thuram was booked for simulation instead.
Thuram still received a standing ovation from the home crowd when replaced, having just struck a thunderous shot which bounced back off the underside of the crossbar, and Inter had little trouble seeing the game out.
Dembélé was left with a simple tap-in.
Alisson made important saves from Barcola and Dembélé in the first half.
His second, from an Andy Robertson free-kick, beat Donnarumma but struck the inside of a post.
Donnarumma was forced to take evasive action repeatedly as Liverpool exerted constant pressure in the closing stages of normal time.