An absorbing contest soon descended into absurdity, though, as VAR - being used for just the sixth time in English club football - disallowed a West Brom goal before awarding Liverpool a penalty moments later, although Firmino hit the crossbar with his spot kick.
West Brom went on to add a third before half time when Joel Matip turned the ball into his own net - a goal which was allowed to stand by VAR - and that proved to be the winner as West Brom held on against late Liverpool pressure.
Mohamed Salah did give Liverpool some hope with his 25th goal of the season 12 minutes from time, but the Reds could not force a replay as they fell to their first home defeat of the season.
The Premier League meeting between these two sides at Anfield last month ended goalless, but it took only six minutes for the deadlock to be broken this evening following a defensive mix-up at the back for West Brom.
Jonny Evans - who has been heavily linked with a move away from the Baggies this month - left the ball for Ben Foster, but the keeper was not coming to claim it and that allowed Salah to race clean through on goal. Foster reacted quickly to thwart the Liverpool top scorer, but the ball bounced back to Firmino, who produced an exquisite first-time chip over the stranded keeper and into the empty net.
Liverpool's lead lasted just 68 seconds, though, as West Brom responded immediately with a high-quality goal of their own when Rodriguez took the ball in his stride before blasting an unstoppable strike into the top corner from the edge of the box.
The hosts were soon on the front foot again as the quick end-to-end nature of the match continued, but Sadio Mane could not get on the end of Salah's cross after good work from Firmino.
West Brom had scored only 25 goals all season before today's visit to Anfield, but they made it two in four minutes to turn the game on its head as a low cross found its way through to Rodriguez, who made no mistake with his finish to double his personal tally for the evening.
Incredibly, three goals in the opening 11 minutes was not the main talking point of the first half, though, and the real drama began in the 19th minute when Craig Dawson appeared to have extended West Brom's lead with a towering header following some shambolic Liverpool defending from a corner.
West Brom's players were left incensed by the decision, and their mood was not improved eight minutes later when VAR came into play once again.
Salah went down under a tug from Jake Livermore inside the area and, while the referee initially ignored the penalty appeals, once again Marriner took a second look. This time it took more than three minutes for the decision to be referred, including Pawson having a second look himself on a pitchside monitor, before a penalty was eventually awarded.
West Brom were once again left furious by the delayed decision having seen VAR take them from 3-1 up to the prospect of being pegged back to 2-2, but Firmino crashed the spot kick against the crossbar to add to the drama.
Still Liverpool poured forward looking for an equaliser, but Evans made a crucial last-ditch block to deny Firmino from inside the area before Mane failed to properly connect with the resulting corner.
Trent Alexander-Arnold then stung the palms of Foster from outside the area before West Brom were hit with a double blow in quick succession, with Kieran Gibbs and Hal Robson-Kanu both being forced off early through injury.
Salah was the next to come close for the home side when he raced down the right channel before drilling a low strike into the side-netting, but it was West Brom that got the game's fourth goal against the run of play.
The VAR controversy led to four minutes of stoppage time, and in the second of those Dawson found space down the right channel before drilling a low cross into the box which Matip glanced into his own net when trying to flick it away from danger.
Once again the celebrations were put on hold as the referee went to the VAR, but this time the decision went the way of the Baggies and the goal was allowed to stand.
That brought an end to an eventful, action-packed and at times farcical half of football, with West Brom becoming the first visiting team to score three first-half goals at Anfield since Real Madrid in October 2014.
It was also the first time that Liverpool had conceded three goals in consecutive home games since January 2007, but their 4-3 triumph over Manchester City in their most recent outing at Anfield would have felt like a long time ago as West Brom started the second half on top too.
Mignolet was called into action to deny Matt Phillips in the opening minutes before the Baggies had the ball in the back of the net for a fifth time, although this time VAR was not needed to rule Dawson offside after he had beaten the Liverpool keeper in the air.
The second half was not as eventful as the first, but West Brom continued to look the most likely to score and Phillips threatened again shortly after the hour mark following an innovative piece of play from Chris Brunt in the buildup.
Liverpool did start to push for a route back into the match in the final 20 minutes and Foster was needed twice in quick succession to deny James Milner and Danny Ings, who had both come on as part of a triple substitution from Jurgen Klopp.
The Reds did pull a goal back with 12 minutes remaining when Firmino teed the ball up for Salah, who calmly stroked his finish into the bottom corner to reach his quarter-century of goals for the club, becoming the fastest Liverpool player to hit that milestone since 1915.
There was to be no comeback for the home side, though, as West Brom held out in the closing stages despite both Livermore and Evans soldiering on with injuries. Virgil van Dijk had the final chance to rescue a replay for the Reds, but his glancing header in the final minute of stoppage time was kept out by Foster.
The defeat means that Liverpool have now lost back-to-back matches for the first time in a year following on from another shock loss to Swansea City on Monday night, while their 19-match unbeaten streak at Anfield also comes to an end.
West Brom, by contrast, have now won three and lost none of their last four games following a 21-match winless run, and today's victory is just their second in their last 13 away games across all competitions.
LIVERPOOL (4-3-3): Mignolet; Alexander-Arnold, Matip, Van Dijk, Moreno; Oxlade-Chamberlain (Ings, 65'), Can (Milner, 65'), Wijnaldum; Salah, Firmino, Mane (Henderson, 65')
WEST BROM (4-4-2): Foster; Nyom, Dawson, Evans, Gibbs (Hegazi, 37'); Livermore, Barry (Yacob, 71'), Krychowiak, Brunt; Robson-Kanu (Phillips, 39'), Rodriguez