Although Arne Slot's side made a dominant start to proceedings, West Ham took the lead thanks to a scrappy own goal inadvertently bundled over the line by Jarrell Quansah, but their advantage was short-lived, as Diogo Jota reacted quickest to head home an equaliser just four minutes later.
Jota then scored his and Liverpool's second with a composed finish four minutes after the interval, before his replacement off the substitutes' bench Mohamed Salah fired a third into the roof of the net with 16 minutes remaining.
West Ham, who caused Liverpool problems throughout the second half, either from set-pieces or on the counter-attack, were left enraged after having two appeals for handball inside the penalty area rightly waved away by referee Andy Madley, either side of Max Kilman squandering a glorious chance to equalise from close range.
Julen Lopetegui appeared to pull a muscle as he jumped in frustration on the touchline, and the Hammers' evening went from bad to worse when Edson Alvarez received a second yellow card for a late challenge on Salah.
Liverpool then rubbed salt into West Ham's wounds with two late goals from Cody Gakpo, securing their place in the fourth round draw that will take place later this evening.
LIVERPOOL VS. WEST HAM HIGHLIGHTS
Jarell Quansah own goal vs. Liverpool (21st min, Liverpool 0-1 West Ham)
Just three minute after Danny Ings had a goal disallowed for offside, West Ham took the lead in scrappy fashion. Max Kilman won the first ball from a corner and knocked it down to Edson Alvarez, who could not adjust his feet inside the six-yard box, but it was Wataru Endo who eventually kicked the ball off teammate Jarell Quansah and it deflected into his own net.
Diogo Jota goal vs. West Ham (25th min, Liverpool 1-1 West Ham)
Liverpool needed just four minutes to respond, with Cody Gakpo's chipped cross from the left acrobatically hooked towards goal by Federico Chiesa, and Diogo Jota reacted quickest to beat Lukasz Fabianski in the air and head home from just a few yards out.
Diogo Jota goal vs. West Ham (49th min, Liverpool 2-1 West Ham)
Curtis Jones deserves plenty of plaudits for his involvement in Liverpool's second goal. The Reds midfielder started the move with a driving run forward and played a one-two with Gakpo before slipping a perfectly-weighted through-ball in behind for Jota to stroke a composed finish into the bottom corner.
Mohamed Salah goal vs. West Ham (74th min, Liverpool 3-1 West Ham)
Mohamed Salah, who came on for Jota just before the hour mark, was the man on the spot to fire a fist-time finish into the roof of the net after Fabianski has parried a powerful close-range shot from Alexis Mac Allister.
Edson Alvarez sent off for West Ham (76th min, Liverpool 3-1 West Ham)
Edson Alvarez received two bookings in the space of just nine minutes, with his second resulting in a red card after catching Salah on the counter-attack with a late a sliding tackle down Liverpool's right flank.
Cody Gakpo goal vs. West Ham (90th min, Liverpool 4-1 West Ham)
Gakpo linked up with Darwin Nunez on the left and cut inside before firing a right-footed shot from inside the area into the bottom left-corner, beating Fabianski at his near post.
Cody Gakpo goal vs. West Ham (90th min + 3, Liverpool 5-1 West Ham)
Gakpo turned West Ham substitute Mohammed Kudus far too easily on the left before looking up and hitting a deflected strike into the far corner - his second of the night and Liverpool's fifth to cap a thumping victory for Slot's side.
MAN OF THE MATCH - CODY GAKPO
Diogo Jota deserves a special mention for his clinical 59-minute display, but Cody Gakpo was a standout performer for Liverpool and was rewarded with two late goals.
The Netherlands international posed a constant threat to West Ham's backline, both creating chances for his teammates down the left and cutting inside to fir shots on goal. On another night, he would have been walking away as a hat-trick hero with the match ball.
LIVERPOOL VS. WEST HAM MATCH STATS
Possession: Liverpool 62%-38% West Ham
Shots: Liverpool 21-12 West Ham
Shots on target: Liverpool 11-3 West Ham
Corners: Liverpool 5-4 West Ham
Fouls: Liverpool 13-10 West Ham
BEST STATS
WHAT NEXT?
Liverpool will now turn their attention back to the Premier League and are scheduled to face Wolverhampton Wanderers at Molineux on Saturday evening.
As for West Ham, they are in action a few hours earlier when they take on Brentford at the Gtech Community Stadium.