Trailing 3-0 from the first-leg City manager Pep Guardiola had said his team needed to produce a "perfect game" if they were to achieve a famous comeback and for 45 minutes they were not far off that level.
The atmosphere was slower to build than for the first leg - and the team arrivals were certainly more low-key - but City fans arrived in hope.
"We played for 10 months amazingly", he added.
The referee was unmoved, though, and with the Dutchman out of position, Fernandinho's through ball found Sterling and his low cross was swept home by Jesus.
The tempo was fierce and City did not relent, with wide men Sane and Bernardo Silva seeing plenty of possession.
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But, we didn't get that one partnership. "We were 50-1 and me and Sanju chose to back our intent, but we lost too many wickets". All of these players are top-notch batsmen, and they can decimate any bowling attack in the world.
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Kevin De Bruyne fired over the bar and Sane had a shot deflected wide.
His goal may have been helped by a foul on Virgil van Dyke in the build-up but it rocked Liverpool back on their heels and whether Klopp intended it or not, his players settled in very nervously for a night under the cosh. The German tapped the ball into the net but the flag was raised, incorrectly, for offside.
For all their possession and excellent attacking football, Guardiola's City continue to find themselves caught out shortly after the interval. Consequently the City boss was ordered to watch the remainder of the game elsewhere by Antonio Mateu Lahoz, a referee with whom he has history.
Four of City's seven defeats this season have come in the competition, with the two by Liverpool following losses at the hands of Basel and Shakhtar Donetsk, while Barca collapsed in dramatic style at the Stadio Olimpico.
Guardiola defended a policy that has seen him spend some £450 million since becoming manager in 2016, but indicated that more would be needed. Barcelona struggled under a barrage of crosses and set-pieces, with first-leg own-goal villains Daniele De Rossi and Kostas Manolas completing the job from a penalty and a corner respectively. Aguero came off the bench but to no avail.
City's awful week after also blowing the chance to seal the Premier League title against local rivals Manchester United at the weekend was rounded off 13 minutes from time when Otamendi was caught in possession and Firmino slotted in off the far post.