Jay Rodriguez's 54th-minute effort looked to be sending West Brom - who were under the leadership of first-team coach Darren Moore - on their way to their first three points since January, but Tammy Abraham levelled for Swansea 15 minutes from time.
The result has left West Brom rooted to the bottom of the Premier League table on 21 points, while Swansea sit in 15th - four points outside of the bottom three.
It was Swansea that had the first half-chance of the match in the 10th minute, but Andre Ayew's effort from distance was always rising over the crossbar.
A tight opening 25 minutes produced few chances for either team, with West Brom's only sight of goal arriving when Chris Brunt's long-range strike was saved by Lukasz Fabianski between the away side's sticks.
The clearest chance of the first period fell to Swansea in the 40th minute when Sam Clucas, who passed a fitness test ahead of the match, found Ayew with a clever pass, but the attacker missed the post.
West Brom took the lead in the 54th minute, however, when Salomon Rondon's flick found Rodriguez at the far post, and the attacker made no mistake from close range.
The Baggies pressed for a second in the period that followed, but Swansea managed to level in the 75th minute when Abraham headed a Clucas cross into the back of the net.
The visitors were the team looking for a winner in the latter stages, but the points were ultimately shared as West Brom stopped a run of eight straight Premier League defeats.
WEST BROM (4-4-2): Foster; Nyom, Dawson, Hegazy, Gibbs; McClean; Brunt, Livermore, Phillips; Rondon, Rodriguez (Krychowiak 83')
SWANSEA (5-4-1): Fabianski; Naughton, Van der Hoorn (Bartley 46'), Fernandez, Mawson, Olsson; A.Ayew (Ki 85'), King (Dyer 61'), Carroll, Clucas; Abraham