Liverpool boss Jurgen Klopp played a youthful side in the third-round match and subsequent replay against Exeter, but included a mixture of first-team players and youngsters for the visit of the Hammers.
Goalkeeper Simon Mignolet, 27, was the oldest player in the Reds' starting line-up, which also included four players - Brad Smith, Cameron Brannagan, Kevin Stewart and Joao Teixeira - who had never started a Premier League match.
This was the ninth game that Liverpool had played in January, and they desperately did not want another match, but that is exactly what they got.
It means their schedule in February will now include at least seven matches, including the League Cup final, two Europa League matches, three Premier League games and one, maybe two, FA Cup ties.
Nevertheless, Klopp will feel his side should have won. A poor touch when only six yards out to waste a chance summed up another ineffective performance from an out-of-form Christian Benteke.
Liverpool were the better team in the second half and created numerous chances, but stand-in captain Allen shot harmlessly over, and Brannagan, Teixeira and Benteke had shots saved by Randolph.
How did Liverpool's youngsters do?
Generally, they were very impressive. Brannagan, 19, who consistently impressed with his set-piece delivery, nearly opened the scoring with a powerful shot from 20 yards out, but goalkeeper Randolph did well to push the ball around the post.
Brad Smith, 21, produced an energetic performance at left-back, often getting forward to help his side's attacks, while handling the threat from the quick Victor Moses with maturity.
Kevin Stewart, 22, a tough-tackling central midfielder, worked hard, making six tackles and two interceptions, and 23-year-old Joao Teixeira, playing on the left side of Liverpool's front three, was twice denied by a diving Randolph.
A wasted opportunity for West Ham?
Three-time winners West Ham were aiming to reach the fifth round of the FA Cup for the fourth time in the last eight seasons - and will have been buoyed when they saw the Liverpool line-up.
They themselves made five changes, although fielded a team rich in Premier League experience apart from Randolph, who played because regular goalkeeper Adrian was back in his native Spain for family reasons.