Jay Rodriguez broke the deadlock in the 54th minute for bottom club West Brom, who were playing under caretaker manager Darren Moore after Alan Pardew left by mutual consent on Monday.
However, the Baggies winless run was extended to 10 league games when Abraham met Sam Clucas corner (75), in what was the forwards first league goal since October.
A first point in nine games leaves West Brom 10 points from safety, while Swansea remain in 15th, four points above 18th-placed Southampton, who are in action at Arsenal on Sunday.
West Brom controlled play in the first half, but their familiar struggles in the final third continued - a Chris Brunt attempt from distance was the closest either side came in the opening 30 minutes.
Andre Ayew should have broken the deadlock for Swansea, but the striker could only stab an attempt wide after latching onto Clucas through ball.
Moments before the half-time whistle, Craig Dawsons cushioned header found Rodriguez, and it took a fine flying save from Lukasz Fabianski to deny the West Brom forward.
West Brom remained on the front foot in the second half, and they took the lead when Rodriguez tapped in after Salomon Rondon flicked on Matt Phillips cross.
The hosts pushed for a second, but Swansea captain Federico Fernandez was on hand to block James McCleans strike from inside the box.
West Brom then gave away a needless corner, and duly paid for it, with Abraham meeting Clucas cross - seemingly with his shoulder - to earn a point for the Swans.
