The Premier League side had not lost any of their last 19 FA Cup games against teams from the fourth tier or lower, but they were pushed hard in a competitive clash at a packed Home Park.
Jurgen Klopp's men earned an early lead through an unexpected source as Lucas headed home a left-wing corner from compatriot Philippe Coutinho.
But the visitors were unable to convert their domination of the ball into further goals despite Daniel Sturridge, Coutinho and Divock Origi all being named in an attacking line-up, the latter seeing a late penalty saved.
Jake Jervis' speculative effort clipped the outside of the post prior to Origi's spot-kick miss and that was the closest Plymouth came to an equaliser.
Klopp's side had to battle hard to book a fourth-round home tie against Wolves, although a fightback from League Two high-flyers Argyle never felt likely.
Sturridge sent an effort over the crossbar and Coutinho mishit a volley as Liverpool controlled proceedings, as they had at Anfield 10 days ago when the sides drew 0-0, in the early stages.
Plymouth were undone after just 18 minutes when Coutinho delivered a left-wing corner and Lucas stole in front of marker Paul Garita to thump in a near-post header, the Brazilian netting for the first time since September 2010.
The hosts almost responded immediately, though, with Liverpool goalkeeper Loris Karius called into action to palm away a swerving Graham Carey drive.
Plymouth went close again after 32 minutes when Oscar Threlkeld got in behind Alberto Moreno to cross, but Trent Alexander-Arnold made a fine challenge to deny Garita a shot on goal.
Origi shot straight at Luke McCormick after 58 minutes as Liverpool struggled for cohesion, with the fit-again Coutinho guilty of wasting possession in good areas on multiple occasions.
A freak second goal almost arrived after 67 minutes as substitute Harry Wilson charged down a McCormick clearance, but the ball flew over to deny the 19-year-old a goal on his Liverpool debut.
Plymouth hit the woodwork after 75 minutes, with Jervis acrobatically turning the ball goalwards with a sweet volley that had Karius beaten.
Liverpool should have made the victory safe with four minutes to go when Yann Songo'o brought down Moreno in the box, but McCormick dived low to his right to keep out Origi's poor penalty.