But, just as the Italians looked to have grabbed a lifeline, Luis Suarez restored the three-goal cushion, by scoring a fourth in the 87th minute.
"It was important for us to win - that's the Champions League".
Ernesto Valverde's side will travel to Rome for the second leg on Tuesday, Edin Dzeko's away goal having given the Italian club the smallest glimmer of hope. "We are happy because it was a complicated match, but they showed they are a good team and we have a long way to go before we can say we are in the semifinals".
"We have to be prepared, we have to forget this first game and play like today - or, if possible, better".
The Giallorossi had 12 shots at Camp Nou, and believed they should have had at least one penalty, but two own goals effectively killed the tie.
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Messi returned to the starting line-up after recovering from his hamstring complaint, as did Sergio Busquets, back after breaking his toe against Chelsea three weeks ago.
City did not manage a single attempt on target in a first half Liverpool completely controlled but came into it more as the hosts, who lost Salah to injury, exhausted in the second period. "Tonight they had help both from the referee and from us".
Dzeko could have won a penalty in the ninth minute when he was pushed over by Nelson Semedo and winger Diego Perotti failed to hit the target from a free header at the start of the second half when they were only a goal down. Pruzzo played as a striker for Roma between 1978-1988, bagging 106 goals in the process.
"It was a penalty on me and the referee has got to have the courage to give it against Barcelona", he told Mediaset.
Valverde was also asked about the future of defender Samuel Umtiti, whose contract talks with Barcelona are yet to reach a positive conclusion. Later, Roma keeper Alisson had to make a stop on Luis Suarez.