The Reds looked set for a goalless draw in the first leg in Italy until Brazil forward Roberto Firmino got on the end of Andrew Robertson’s corner to head his side into the lead.
Mohamed Salah added a second with a deflected effort from inside the box which means Jurgen Klopp’s side take a sizeable advantage into the second leg at Anfield on 8 March.
Italian champions Inter hit the bar through Hakan Calhanoglu and were on top for long periods of the second half before Firmino struck in the 75th minute.
Liverpool scored from their only two attempts on target and will be firm favourites to progress when the two sides meet in the deciding leg next month.
It was a particularly memorable night for midfielder Harvey Elliott, who became the youngest player ever to start a match in the competition for Liverpool at the age of 18 years and 318 days.
The only negative on an otherwise upbeat night for the Reds was an injury to Portugal forward Diogo Jota, who was replaced by goalscorer Firmino at the start of the second half after suffering an ankle injury.
Lethal Reds impress in Italy
Liverpool have now won seven successive matches in all competitions, with this arguably the most impressive of the current run.
But Klopp’s team had to dig deep as Inter looked the more threatening side after a strong start by the Premier League club.
Former Manchester City forward Edin Dzeko found himself clean through in the first half before a vital challenge by Virgil van Dijk, who was captaining the Reds in the absence of Jordan Henderson.
The hosts went close again when Ivan Perisic whipped in a cross, but Lautaro Martinez was unable to give Inter the lead.
In a fascinating match, Liverpool took the lead through Firmino’s glancing 75th-minute header before Salah doubled the lead eight minutes later.
There is still work to do for Klopp’s team but it is going to require something extra special from Inter to turn this tie around.
Liverpool remain in the hunt for four trophies and could have one in the bag by the time Inter visit Merseyside, with the Reds facing Chelsea in the Carabao Cup final at Wembley on 27 February.
Teenager Elliott embraces San Siro stage
Despite it only being half full because of Covid-19 restrictions, there was an incredible atmosphere inside the San Siro for the first meeting between these two clubs for 14 years.
Elliott, who does not turn 19 until April, embraced the occasion on what was his first start since suffering a serious ankle injury in September.
The teenager did not look out of place on one of the most famous stages in world football and produced a cross in the first half which was just cut out before Salah could poke it over the line.
Elliott was replaced in the 59th minute as part of a triple change as Klopp attempted to turn the tide as Inter threatened – a move that worked as Liverpool went on to establish a strong advantage.
“He did the job, he was working incredibly hard, but when you’re 18 years old and playing one of the most experienced teams in world football, it is not a walk in the park,” said Klopp.
“So it was an important game for him and very helpful for us as well.”