The Reds booked their place in the next round as they overcame rivals Everton 2-1 on Friday night with £75 million centre-back Virgil van Dijk marking his debut with an 83rd-minute headed winner.
Alan Pardew's West Brom meanwhile ended a 21-game winless streak at League Two play-off hopefuls Exeter City, winning 2-0 thanks to first-half Salomón Rondón and Jay Rodriguez strikes.
And picked out as the first ball of the draw, Jürgen Klopp's men have been paired to face the relegation-threatened Baggies - 19th in the top-flight table - at Anfield.
The two teams last went head-to-head on December 13 when they drew 0-0 on Merseyside, with Liverpool unbeaten in the last nine meetings between the two clubs.
The tie will be played on the weekend of January 27 and 28, between a trip to Swansea City on Monday 22 and an away clash with Huddersfield Town on Tuesday 30.
The other stand-out fourth-round clashes include Yeovil Town vs Manchester United, Cardiff City or Mansfield Town vs Manchester City and Newport County vs Tottenham Hotspur.
How important is the FA Cup to the Reds?
In their two previous seasons under Klopp, Liverpool have been knocked out in the fourth round on both occasions - losing to West Ham United in extra time in 2015-16 before being humbled at home to Wolves last year.
There will undoubtedly be a desire to change that this season, particularly with it the most likely source of some overdue silverware - Liverpool winning just one trophy in 12 years since their last triumph in this tournament.
Philippe Coutinho's recent big-money departure somewhat alters their hopes of success in the Champions League and the Premier League and so the FA Cup carries huge importance.
Under Klopp alone Liverpool have lost two cup finals, while they have been knocked out in the semi-finals of the FA Cup once and League Cup twice in the last three seasons.
They have reached the final four in England's other domestic cup competition in all of their last three campaigns but failing to do so this year, being knocked out by Leicester City, has freed up their January and afforded them extra rest.
And alongside Klopp's heavy rotation, Liverpool are well placed to compete across three competitions and so will likely treat this competition very seriously throughout - having named a strong line-up to face Everton.
So they should, as they have a very real chance of going on to win it. Arsenal are already out of the competition, and though all the rest of the Premier League's 'big six' are in the fourth round, they also have other commitments - with all bar Liverpool and Spurs in the semis of the League Cup.
At the very least Klopp will be confident of his side progressing through to the fifth round for the first time in his tenure as seven-time FA Cup winners Liverpool look to move one step closer to a showpiece occasion at Wembley Stadium.