The win means Liverpool will now face Chelsea in the fifth round at Stamford Bridge.
Reds manager Jurgen Klopp nor any of his senior players were in action with Under-23s boss Neil Critchley leading out a squad with an average age of 19 years and 102 days.
It was a quiet opening 20 minutes to the game but the tie opened up when a mistake from Ki-Jana Hoever allowed Shrewsbury to get in behind, only for Pedro Chirivella to step in and defend.
Liverpool were on top for the remainder of the half and Curtis Jones fired a free-kick straight at Shrewsbury goalkeeper Max O’Leary before creating a good opening following a passing combination with Harvey Elliott.
The hosts were on top at the start of the second half, too, but it was Shrewsbury who thought they had taken the lead after 59 minutes.
David Edwards smashed Daniel Udoh’s cross at Caoimhin Kelleher, who could only palm upwards for Shaun Whalley to head home, but VAR overturned the effort for offside.
Liverpool responded with more pressure – including by racking up a double figure of second-half corners – and the pressure eventually told after 75 minutes.
Neco Williams’ cross was sent in and there was a mix-up between O’Leary and Shrewsbury centre-back Ro-Shaun Williams.
O’Leary came off his line to collect the cross but Williams attempting to head it back to his keeper from the edge of the area only for the ball to loop over the hapless O’Leary and in.
Shrewsbury tried to pile on the pressure in the final few minutes but Liverpool got through.
Also on Tuesday, Wayne Rooney scored to help Derby County beat Northampton Town 4-2 in the FA Cup fourth round.
The win means Rooney will face former club Manchester United in the fifth round.
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