The Reds, who eased through 6-1 on aggregate, held off Porto in the early stages before Salah and Sadio Mane combined for the Senegalese to put them ahead with their first shot on target following a VAR review.
Egypt striker Salah doubled the tally after the break and goals from Roberto Firmino and Virgil van Dijk wrapped it up late on.
Porto had briefly reduced the arrears through Eder Militao, but having lost the first leg 2-0 at Anfield, they never seemed in a position to overturn the deficit.
Five-times European champions Liverpool, who had produced a scintillating display to beat Porto 5-0 in the last-16 at the Dragao stadium last season, had a totally different approach this time.
They soaked up the pressure as Porto made an adventurous start, with Jesus Corona curling a powerful shot just over the bar in the opening minute.
Juergen Klopp’s side looked temporarily toothless, but they were just waiting for their moment - which came in the 26th minute.
Mane latched onto a cross from Salah to beat Iker Casillas from close range but he was initially flagged offside, only for referee Danny Makkelie to check the VAR.
After a two-minute review, the goal was eventually given.
Facing the huge task of scoring four goals to qualify, Porto looked stunned, despite having attempted 15 shots before the interval.
Salah, who endured a goal drought in March, confirmed he was getting close to his best again when he coolly slotted the ball past Casillas after collecting a perfect diagonal through ball from Trent Alexander-Arnold in the 65th minute.
It was Salah’s first Champions League goal since December, and his first away from Anfield in the competition in 12 months.
Militao soothed the home fans’ suffering a little, beating Roberto Firmino in the air to head home in the 69th minute.
Brazilian Firmino, who had started on the bench as Klopp handed a rare start to Divock Origi, made it 3-1 with a downward header with 13 minutes left and Van Dijk headed home a fourth from a corner eight minutes later.
Spurs stun City after seven-goal thriller, VAR drama
Meanwhile, Tottenham Hotspur reached the semi-finals on away goals after an astonishing encounter at the Etihad Stadium, despite a 4-3 defeat to hosts Manchester City.
Tottenham's Fernando Llorente celebrates scoring their third goal with team mates - Champions League Quarter Final Second Leg - Manchester City v Tottenham Hotspur - Etihad Stadium, Manchester, Britain - April 17, 2019. (Photo: Reuters)
Spurs, who won the first leg 1-0 and progressed after the tie finished 4-4 on aggregate, will play Ajax Amsterdam for a place in the final against Liverpool or Barcelona after one of the most remarkable matches in the competition’s history.
It started in manic fashion with four goals in the opening 11 minutes of the game. It was a gripping spectacle right up until the dramatic finale when City had a stoppage time goal from Raheem Sterling, which would have sent them through, ruled out for offside after a VAR review.
City fans had celebrated what looked like the goal that sent them into the last four before the replay confirmed that Sergio Aguero was offside in the build-up.
The contest exploded into life in the fourth minute when Sterling put City in front, receiving a ball from Kevin De Bruyne on the left flank and then cutting inside onto his right foot before beating Hugo Lloris with a beautiful, curling shot into the far corner.
Three minutes later though, Spurs drew level when City’s French defender Aymeric Laporte’s attempted clearance fell straight at the feet of Son Heung-min whose low shot flew in off the leg of City keeper Ederson.
Laporte was also at fault for Spurs’ second in the 10th minute when he gifted the ball to Lucas Moura, the Brazilian feeding Christian Eriksen who found Son, the brilliant Korean finishing superbly into the far corner.
There was no respite as City responded immediately as Aguero fed Bernardo Silva and the Portuguese winger’s left-foot shot deflected off Danny Rose and beat the wrong-footed Lloris to make it 2-2 on the night.
The dangerous Sterling struck again in the 21st minute to level the aggregate score, ghosting in at the back post to convert a low cross from De Bruyne that had zipped across the face of the goal.
A thundering strike from Aguero in the 59th minute put City in the driving seat again - the Argentine blasting in at Lloris’s near post after a magnificent burst forward and perfectly weighted and timed pass from De Bruyne.
City, who went out at the same stage to Liverpool last season, were stunned though in the 73rd minute when Spurs substitute Fernando Llorente bundled in a Kieran Trippier corner to restore the London club’s advantage through away goals.
With officials unsure if the ball had bounced off the Spaniard’s hand or rolled off his hip in the goalmouth scramble, VAR was called into action and the goal stood for Spurs.
Then came the stoppage time video torture for the home support - Aguero burst down the right flank and delivered a low cross for Sterling to slot home what would have been his hat-trick goal.
The VAR call was correct and the Premier League champions were out - to the delight of Tottenham manager Mauricio Pochettino and his players who celebrated wildly in front of the travelling Spurs support.
