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.
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.
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And the hosts' cause was further damaged when defender Billy Jones was withdrawn through injury and replaced by Victor Anichebe.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.
