Battered by constant Manchester United pressure in one of the most one-sided first halves at Old Trafford this season, Albion rolled up their sleeves and battled back into the game after the interval.
With only two points separating bottom club West Brom from 17th place Norwich, Albion must beat Portsmouth at The Hawthorns next Sunday to have a chance of staying up.
Despite being without Ruud van Nistelrooy and having Wayne Rooney and Louis Saha on the substitutes bench, United tore at the Baggies from the start.
Albion's last win at Old Trafford was in 1978 and they could have been behind in the 10th minute.
Mikael Silvestre picked out Cristiano Ronaldo who beat Albion's former United defender Ronnie Wallwork and fired in a shot which goalkeeper Russell Hoult saved at full stretch.
Hoult turned over a fierce 25-yard shot from Quinton Fortune four minutes later and from the ensuing Ryan Giggs corner, Alan Smith messed up a free header from only six yards out.
Hoult again had to be alert soon after to turn away Neil Clement's attempt to cut out Ronaldo's superb cross from the right.
That save was to prove a key point in the game. Hoult looked to have pulled a muscle in his groin making it.
When Thomas Gaardsoe fouled Smith two yards out on the left side of the Albion box, Giggs' quickly taken free-kick took the defenders by surprise and Hoult was unable to get across goal to keep the effort out of the bottom corner.
Seconds later, Hoult limped off to be replaced by Tomasz Kuszczak, who had only one senior Albion appearance to his name, and he was to prove the Albion saviour with a string of magnificent saves.
After Giggs had fired into the sidenetting and Smith headed wide, Kuszczak's first real contribution to the game almost caused disaster.
As he ran to meet Giggs down the left when the Welshman had been sent clear by John O'Shea's pass, Giggs rounded him and sent a shot across goal which passed inches from the line.
In the 37th minute, Wes Brown found Ronaldo and his raking cross from the right was met by the diving Fortune at the far post.
But the South African's header was brilliantly saved by Kuszczak and, seconds later, Albion's reserve goalkeeper did well to turn over Ronaldo's snap shot from 12 yards.
Kuszczak made a fine double save from Ronaldo soon after and United defender Rio Ferdinand, reputed to be wanting £120,000 a week to sign in a new contract, earned the derision of the crowd in the dying seconds of the half.
Unmarked on the right side of the Albion box he fell over the ball when well placed to get in a shot, bringing cries of "100 grand a week, you're having a laugh" from the crowd.
Kuszczak again denied United three minutes after the break.
Giggs raced past Martin Albrechtsen and teed the ball up for Kleberson to fire in a 20-yard shot which Kuszczak did well to turn over the bar.
The swirling wind which aided United in the first half was to help Albion in the second.
O'Shea misjudged his 50th minute header back to Roy Carroll. The ball hung in the air before landing perfectly for Kevin Campbell inside the United box and only the United goalkeeper's speed of his line to clear off Campbell's head saved the day.
Kleberson fired over when well placed and United went close again on the hour when Phil Neville's long pass set up Ronaldo, who fired inches wide of the post.
Albion were now starting to cause United problems and equalised in the 62nd minute.
As O'Shea and Geoff Horsfield fought for the ball inside the United box the Albion striker went down.
It looked innocuous, but referee Mark Halsey pointed to the spot and Robert Earnshaw coolly stepped up to send Carroll to his left, while clipping the resulting penalty to his right and into the back of the net.
Six minutes later United brought on the heavy brigade when Rooney, Saha and Paul Scholes replaced Kleberson, Smith and Neville respectively.
It sparked a United revival.
Kuszczak saved well from Rooney and Albion were lucky to survive in the 86th minute.
Ronaldo rolled a free-kick from the right across the line to Scholes, whose crisp low shot from 25 yards crashed against the foot of a post.
Kuszczak turned behind another firm drive from Rooney in the last minute, but Albion hung on to fight another day.
Man of the Match: Tomasz Kuszczak - Replaced the injured Russell Hoult after just 22 minutes and produced a series of magnificent saves to earn his side a possibly crucial point.