Lautaro Martinez scored the only goal of the game midway through the second half with a brilliant long range strike.
Inter were reduced to 10-men minutes later, when Alexis Sanchez received a second yellow card for a high challenge on Fabinho.
Jurgen Klopp said: ""Without playing brilliantly, we had a lot of clear-cut chances. They didn't have that many, but they scored."
The defeat was the Reds' first in over a year at Anfield and ended their 12 game winning run in all competitions.
Story of the game
The visitors made a positive start to the match, with Ivan Perisic and Sanchez looking the most likely to threaten the Liverpool goal.
Klopp’s side struggled to gain any real control of the game in the opening 20 minutes due to Inter’s intensity out of possession, much like the first leg at the San Siro.
A medical incident in the Kop halted play for several minutes midway through the first half, although medical staff were fast on the scene and thankfully took control of the situation.
With half an hour played, the Reds were inches away from taking the lead. A Trent Alexander-Arnold free kick was guided towards goal by Joel Matip, but the Cameroonian’s header bounced off the crossbar and back into play.
Moments later, an Alexander-Arnold corner looked certain to be nodded home by Virgil Van Dijk, but Milan Skriniar threw his body at the ball to prevent the dutchman putting the tie to bed.
Five minutes before the break, Hakan Calhanoglu fired a venomous low free kick at goal, however Alisson reacted well to push the ball to safety and away from the onrushing Inter forwards.
With the final kick of the half the Reds thought they'd confirmed a place in the quarter-finals, but Alexander-Arnold's 25 yard free kick curled just wide of Samir Handanovic’s goal, after Arturo Vidal was cautioned for a foul on Curtis Jones.
The second half began at a fast pace, with both Sanchez and Alexander-Arnold having efforts on goal in the opening exchanges.
In the 52nd minute, LFC should've taken the lead. A clever dinked pass from Thiago found Diogo Jota, who got a toe to the loose ball but couldn't loop over the onrushing Handanovic. The ball dropped to Mohamed Salah, who to the Anfield faithful's surprise didn't hit the target from 12 yards as his effort bounced back off the post.
The match then began to get stretched, with the visitors having more joy on the counterattack. A switched ball from Vidal found Perisic, who pulled a cross back to Martinez in the box. The Argentine shifted the ball but didn't connect with his effort cleanly, as it rolled wide of the target.
Salah was made to rue his missed opportunity with half an hour to play, as Inter netted the opener. The visitors won the ball back in LFC's half, and Sanchez found Martinez who struck a thunderous swerving effort into the top right corner of Alisson's goal.
However, the away side's hopes of any momentum were dashed moments later as they were reduced to 10-men. Fabinho latched onto a loose ball in front of Sanchez, who flew in dangerously and caught the Brazilian. Referee Antonio Mateu reached for a second yellow for the ex-Arsenal man, and made the Italians task all the more difficult.
Jordan Henderson and Naby Keita were introduced soon after in an attempt to gain some control of proceedings, as Simone Inzhagi made a bulk of changes of his own in an attempt to take the game to extra time.
Klopp's side again looked primed to settle the tie with 15 minutes to go. A lofted cross from Sadio Mane found Salah, who only had the goalkeeper to beat from 10 yards. The Egyptian struck the ball first time on his right foot and his effort crashed painfully off of the Slovenians right-hand post.
The closing stages were far from comfortable for the Reds, but they had the chance to seal their fate in the 92nd minute. A quick counterattack saw Alexander-Arnold cross to Luis Diaz, who controlled the ball and looked certain to slot home from six yards. However, Vidal made a last ditched effort to get a brilliant block over the crossbar.
The match drew to a close after 96 minutes, as Klopp's side edged through to the last eight.