In an entertaining encounter, three goals in a manic opening eight minutes set the tone as Liam Millar netted either side of a leveller from Pools skipper Jacob Owen to put Liverpool in command.
Glen McAuley extended the Reds’ advantage as the half drew to a close while Millar was able to complete his hat-trick after the interval.
Curtis Jones then added a fifth late on, but Pools can be proud of a display that gave their Premier League opposition plenty to think about.
Ian McGuckin kept faith with the same side that had defeated Carlisle United in the second round as Pools looked to contain a Liverpool side packed with attacking quality.
And that quality was clear to see with just two minutes on the clock as the visitors took the lead - Millar slotting the ball under keeper Alex Lawrence after getting on to a fine through ball.
But Pools rallied and were back on level terms shortly after when a free-kick was nodded across goal by Connor Simpson, with Owen pouncing to force the ball home from close range.
Parity lasted just two minutes though as the Reds regained the lead through the lively Millar, as he twisted and turned in the area before drilling into the bottom corner.
Liverpool continued to look a threat with McAuley firing into the side netting moments before sending a close range header into the arms of Lawrence.
The frontman wasn’t as wasteful with his next chance though, as he finished well past Lawrence to give his side a two-goal cushion at the break.
Gerrard’s side dominated possession after the interval but struggled to break down Pools’ well-organised backline - although Jones and Yasser Larouci did force stopper Lawrence into smart stops.
But as tiredness crept in the Reds took full advantage with Millar securing the match ball with a well-taken effort from an acute angle.
There was to be a fifth too, with Jones cutting inside and riding several challenges before unleashing an unstoppable effort into the top corner to seal Liverpool’s place in the fourth round.
Pools XI: Lawrence, Holliday, Cooper, Bayne, Owen, Miller, Pritchard (Short 80), Orrell, Simpson, Weirs (Robinson 80), Boyle (Jackson 64)
Subs Not Used: Stephenson, Murphy
Goals: Owen (6)
Bookings: Boyle (18), Pritchard (56)
Liverpool Xl: Jaros, Larouci, Boyes (Raitanen 69), Williams, Glennon, Clayton, Tagseth, Dixon-Bonner, Millar (Simmonds 75), Jones, McAuley (Longstaff 59)
Subs Not Used: Sharif, Atherton
Goals: Millar (2, 8, 72), McAuley (37), Jones (77)