Boulaye Dia handed Unai Emery's men an early lead in front of a boisterous home crowd, before Francis Coquelin stunned the below-par visitor by wiping out its aggregate lead on the stroke of half-time.
But Liverpool grew into the game after its dismal start, and after Geronimo Rulli failed to make a routine stop from Fabinho's effort, half-time substitute Diaz headed home to send Jurgen Klopp's men to the final.
Sadio Mane raced clear to round Rulli and roll home a late third to make the result safe before Etienne Capoue was sent off late on, keeping the Reds on course to cap an incredible season by winning four major trophies.
After failing to record a single shot on target at Anfield, the Yellow Submarine needed just three minutes to open the scoring, Dia tapping home after Capoue turned Pervis Estupinan's delivery across goal.
Gerard Moreno saw a close-range header blocked as the visitors produced a dreadful first-half performance, and the Reds' advantage, which looked to be decisive prior to kick-off, was wiped out when Coquelin sparked wild scenes by heading Capoue's cross into the top-left corner.
Trent Alexander-Arnold struck the top of the crossbar with a deflected effort as Liverpool improved after the break, before Fabinho drilled a low shot through the legs of Rulli to restore the visitors' aggregate lead after 62 minutes.
Diaz went close to bending home a superb second moments later, but was on hand to nod home Alexander-Arnold's cross after 67 minutes and put the Reds back in full command of the tie.
The tie was settled once and for all when Mane took advantage of another Rulli error after 74 minutes, rounding the keeper well outside his area before rolling home to secure Liverpool's progress, with Capoue then dismissed for a second yellow card after fouling Curtis Jones.