With Juergen Klopp opting to rest his senior squad for the replay during the Premier League's winter break, Milner, 34, took it upon himself to get heavily involved in the young side's buildup to the clash at Anfield, which Liverpool won 1-0.
"He trained with us yesterday and he asked whether he could come along. I was like, 'I think the answer is yes to that one. Of course you can,'" He was very respectful to ask," Critchley told reporters.
"He has achieved so much in the game because of who he is; he was giving words of advice to the players, he was getting right behind them, he was vocal in the dressing room and animated behind me."
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Critchley said Milner, who has been sidelined with injury, was delighted with the performance against the third-tier side.
"I don't think he's available yet ... but he was jumping up and down behind me so I think he was desperate to get on," said Critchley, adding that Klopp had also sent a message to the team.
"He's buzzing. We've given him a (fifth round) game against Chelsea away to look forward to." (Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Peter Rutherford )
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