It looked an unlikely outcome after Sadio Mane put Liverpool ahead after only 12 minutes with the visitor forced back into its own half for long periods.
But in the second half Albion offered more of a threat and with eight minutes remaining, Nigerian international Semi Ajayi rose to head home a cross from Matheus Pereira. It was Ajayi’s first Premier League goal.
Liverpool surged forward in search of a late winner but West Brom keeper Sam Johnstone made a stunning fingertip save to keep out Roberto Firmino’s header.
Firmino scored the late winner in the previous home clash against Tottenham Hotspur as Liverpool hit the top, but there was to be no repeat.
Allardyce, who took over from Slaven Bilic at West Brom, has never been in charge of a relegated side. He has a fine record managing teams at Anfield, unbeaten in four games with three different clubs, including the last win by a visiting manager with Crystal Palace in April 2017.
Goalkeeping hero Johnstone acknowledged the difference made by Allardyce since he took charge.
“Everyone knows what the manager is like. He has come in and things change very quickly and the lads are reacting to it.
“It is new management, new things and we keep going and believe in ourselves,” he told Sky Sports.
“West Brom deserved the point tonight because we didn’t finish the game. It is difficult to create. In the end everyone knows and feels we should have and could have done better,” he told Sky Sports.
The result leaves his side three points clear at the top from Merseyside rivals Everton with a host of other clubs in touching distance in the middle of a hectic holiday run of matches.
Liverpool started the match, the only one played before a home crowd with other clubs under coronavirus restrictions, in the mood for a goal glut. Senegalese star Mane capitalized on a period of intense pressure by controlling Joel Matip’s sharp pass on his chest before firing past Johnstone.
Other goals might have followed but the Baggies held firm and Liverpool went off the boil, unable to test Johnstone.
West Brom showed a little more ambition after the break and might have leveled when Karlan Grant got in behind youngster Rhys Williams, a second half substitute for the injured Matip, but Liverpool keeper Alisson Becker denied him with his legs.
But the warning signs were there as Ajayi found space to direct his equalizer off the woodwork and into the net.
Leeds edges past Burnley
Earlier, Leeds cemented its place in mid-table as Patrick Bamford’s penalty settled a 1-0 win over Burnley.
Bamford converted from the spot on five minutes after being brought down by the onrushing Nick Pope, his 10th goal of the season.
Burnley felt hard done by when Ashley Barnes appeared to equalize in the 19th minute. Leeds’ French goalkeeper Illan Meslier came out to claim a high ball but fumbled it after clattering into the back of Burnley’s Ben Mee. Barnes put the dropped ball into the goal but the goal was disallowed as referee Robert Jones called a foul on Mee and awarded a free kick to Leeds.
The visiting players were outraged that it could not be referred to the video assistant referee because Jones had blown for the foul before the ball hit the net.
Meslier, keeping his fifth clean sheet of the season, kept out a late effort from Barnes to preserve three points for Marcelo Bielsa’s men.
Sunday’s other afternoon match saw a 2-2 draw between West Ham and Brighton, with the visitor twice leading at the London Stadium only to be pegged back both times.
Neal Maupay put Brighton deservedly ahead just before halftime but the Hammers equalized on the hour mark through Ben Johnson’s first goal for the club.
Lewis Dunk looked to have won it for Brighton when he converted following a corner, with West Ham appealing for handball, but Tomas Soucek scored another crucial goal, his fourth of the season, when he headed home unmarked from a corner.
The result leaves Brighton languishing two points clear of the relegation zone with West Ham in 10th.
Jose Mourinho’s Tottenham are playing at Wolves in the late kickoff, hoping to return to winning ways after two straight defeats, including the reverse at Liverpool.