Lucas had gone 2,316 days without a goal before heading home during the opening stages, and despite a missed penalty from Divock Origi, the Merseyside outfit were able to hold on to progress past the League Two outfit and into the FA Cup fourth round.
Plymouth did not show much ambition in the final third in the first game at Anfield, but it took just six minutes for Derek Adams's team to threaten as Paul Garita miscued a header from the penalty spot when he should probably have hit the target.
However, it did not take much longer for the visitors to take the lead as Lucas directed a powerful header into the net after meeting Philippe Coutinho's corner at the near post.
Back up the other end, Trent Alexander-Arnold almost found Sturridge for a tap-in from close range, before the young right-back had to be alert to stop Garita scoring from eight yards after excellent work from Threlkeld.
Soon after the restart, David Fox forced Karius into a save from around 20 yards from goal, and while Liverpool continued to offer a threat through Sturridge, Plymouth were the better side as the game hit the hour mark.
Liverpool hit back through substitute Harry Wilson, who blocked a Luke McCormick clearance before seeing the ball go over the crossbar, while Sturridge hit a low shot towards the far corner only to see the goalkeeper get his body in the way.
Plymouth were not finished, however, and they came agonisingly close to an equaliser with 15 minutes left when Jake Jervis met Sonny Bradley's headed pass, only to see his scissor kick from 12 yards strike the post and rebound away from goal.
With four minutes to go, Liverpool were presented with an opportunity to make sure of their place in the fourth round after Yann Songo'o upended Alberto Moreno in the area, but Origi's penalty was poor and McCormick was able to make the save.
The Belgian almost redeemed himself with a fine individual effort which went just over the bar, but it mattered little in the end as Liverpool comfortably held on to move through to the last 32.