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A huge contingent of travelling Shrews fans were dreaming of a massive upset after Daniel Udoh put the League One side ahead at Anfield on Monday (AEDT).
But Liverpool clicked into gear in the first half, Gordon becoming the second-youngest goalscorer in the Premier League giant's history at the age of 17 and 96 days and Fabinho on target from the penalty spot.
Jurgen Klopp, back on the touchline after an absence following his COVID-19 test, went with a mix of youth and experience to see Reds into the fourth round, Roberto Firmino adding a third with a clever backheel and Fabinho putting the icing on the cake right at the end.
Liverpool dominated possession without posing a real threat before Shrewsbury sensationally took the lead after 27 minutes, when an unmarked Udoh converted Nathanael Ogbeta's inviting cross.
Shrewsbury's lead lasted only seven minutes, as the composed Gordon controlled Conor Bradley's pass and applied a left-footed finish from inside the penalty area.
Ryan Bowman found the back of the Liverpool net with a bullet header soon after, but the striker had strayed offside.
Ethan Ebanks-Landell then gifted Fabinho the chance to put Klopp's side in front just before half-time when he handled in the box and the Brazil midfielder sent Marko Marosi the wrong way from 12 yards.
A brilliant save from Marosi prevented full debutant Elijah Dixon-Bonner from extending Liverpool's lead.
However, substitute Firmino took a touch to control Ibrahima Konate's shot before scoring with a cheeky finish 12 minutes from time and Fabinho lashed home late on as the Reds eased into round four.