However, the goal was disallowed in controversial fashion and West Brom held on to come away from Anfield with a point which lifts them out of the relegation zone and back up to 17th place.
Liverpool, meanwhile, remain fifth courtesy of a second home draw in the space of four days having failed to score in a game for the first time since October.
With Burnley's win over Stoke City on Tuesday night seeing Liverpool drop out of the Champions League places, Jurgen Klopp welcomed Philippe Coutinho and Roberto Firmino back into his starting lineup, having rested them for the Merseyside derby at the weekend.
Those were two of six changes made by Klopp from Sunday's draw with Everton, although the front four of Mohamed Salah, Sadio Mane, Coutinho and Firmino struggled to break down a stubborn West Brom defence in the opening exchanges.
Salah and Firmino did combine to create the first clear chance of the game 17 minutes in, and it should have resulted in the opening goal too as Salah's incisive pass found Firmino, who slid his finish inches past the far post from an acute angle.
Claudio Yacob had the visitors' first sight of goal with a header which flew off target five minutes later, and it was the Baggies who came closest to breaking the deadlock in the first half when Hal Robson-Kanu's dipping strike from range clipped the top of the crossbar with Loris Karius beaten.
Coutinho almost came up with an immediate response when he worked space to shoot before firing a 25-yard effort into the arms of Ben Foster, and Salah also threatened the opening goal shortly afterwards but was unable to get anything on the end of Trent Alexander-Arnold's dangerous delivery.
Liverpool continued to knock on the door at the end of the first half, but West Brom had the bodies back in defence to deal with the danger as both Firmino and Salah were denied by blocks in quick succession from inside the penalty area.
It was West Brom who started the second half on top, though, winning a string of set pieces which forced Karius into saves from Grzegorz Krychowiak and Yacob before Ahmed Hegazi put a header well over when he should have done better.
Liverpool soon responded, but once again they failed to capitalise on the rare occasion they cut through the Baggies' defence, with Mane firing one effort wide from a good position before Salah could not react in time to steer Alexander-Arnold's cross on target.
Having frustrated their hosts for most of the match, West Brom almost gifted Liverpool the opening goal when Foster fell victim to a swirling wind inside Anfield, misjudging an attempted punch and then spilling the ball when trying to atone for the error, but Georginio Wijnaldum could not react to the loose ball in time to turn it home from close range.
It was not until the final 10 minutes that Liverpool really started pushing for a winner, though, and they thought they had it in the 82nd minute when Solanke - on as a sub for Mane - bundled the ball into the back of the net after Hegazi had failed to cut out a cross from Gomez.
The goal initially looked to have been given, but referee Paul Tierney belatedly chose to disallow it for a handball, much to the chagrin of Klopp.
Liverpool were immediately back on the front foot, though, and almost made it a meaningless debate two minutes later when Alex Oxlade-Chamberlain was thwarted by Foster and Solanke then saw his follow-up effort cleared off the line by Hegazi.
Oxlade-Chamberlain had another sight of goal moments later having cut inside from Solanke's pass, but he drilled his finish straight at Foster as West Brom continued to hang on for the point which lifts them back out of the bottom three.
The Baggies are still goalless under new manager Alan Pardew, though, while the draw also sees them set a new unwanted club record of 16 consecutive games without a win across all competitions.
Liverpool, by contrast, are now unbeaten in 11 matches but the hosts will be the more disappointed of the two sides with the result having once again failed to make their chances pay at Anfield.
Liverpool (4-3-3): Karius; Alexander-Arnold (Gomez, 80'), Lovren, Klavan, Robertson; Wijnaldum (Oxlade-Chamberlain, 76'), Can, Coutinho; Mane (Solanke, 76'), Salah, Firmino
West Brom (4-3-3): Foster; Nyom, Hegazi, Evans, Gibbs; Livermore, Yacob, Krychowiak (Brunt, 79'); Robson-Kanu (Rodriguez, 71'), Rondon, McClean