Stekelenburg initially pulled off a number of fine saves to keep Fulham ahead, but proved to be the villain of the piece as he allowed substitute Matej Vydra's shot to squirm under him and roll narrowly over the line with four minutes to go.
It had looked as though Magath was going to be indebted to a familiar face in Ashkan Dejagah, who won the Bundesliga title with the German in 2008-09 in his spell at Wolfsburg and put Fulham ahead after 28 minutes.
But the Premier League's bottom club were denied a first win in seven league games when Stekelenburg blotted his copybook in the closing stages.
The result leaves Fulham four points adrift of safety at the foot of the division, while West Brom are a point above the drop zone in 16th, although Pepe Mel is still waiting for his first win since taking over last month.
January signing Konstantinos Mitroglou was named on the bench for Fulham, who started brightly and should have gone ahead when two other new recruits combined as Johnny Heitinga headed over from Lewis Holtby's left-wing cross.
Fulham continued to dictate play without creating too many chances, although Steve Sidwell did blaze over from point-blank range after 21 minutes following good work down the right from Hugo Rodallega.
But for all their pressure, the visitors were thankful to goalkeeper Stekelenburg for keeping them level, the Netherlands international standing firm to deny Thievy Bifouma after the Frenchman had combined brilliantly with compatriot Morgan Amalfitano on the counter-attack.
Magath's men were undeterred, though, and deservedly took the lead just before the half-hour mark when Dejagah collected Kieran Richardson's delivery before powering a right-footed finish beyond Ben Foster.
West Brom struggled to mount a response and would have fallen further behind had Foster not pulled off a fine diving save to keep out Rodallega's long-range drive.
And the hosts' cause was further damaged when defender Billy Jones was withdrawn through injury and replaced by Victor Anichebe.
Mel's men appeared reinvigorated after half-time, though, with Stekelenburg forced into another impressive save to prevent Chris Brunt from levelling matters.
The Hawthorns outfit were dominant throughout the second period but Steklenburg's inspired performance continued as he prevented James Morrison from equalising after 66 minutes with a vicious effort from the edge of the area.
It looked as though West Brom's effort were going to be in vain, but the Midlands club salvaged a point in dramatic fashion when former Roma keeper Stekelenburg somehow let an effort from Vydra, who came off the bench only 11 minutes earlier, roll under his body.
Stekelenburg did redeem himself with a brilliant save to keep out an injury-time strike from Brunt, with Anichebe firing wide on the rebound, missing a gilt-edged chance to complete the turnaround.
