An error by Marcos Alonso led to Callum Robinson opening the scoring for West Brom in the fourth minute and the Irish striker made it two when Chelsea's new signing Thiago Silva, who was given the captain's armband, miscontrolled the ball.
Kyle Bartley was left unmarked to add a third in the 27th minute with a close-range finish to send Chelsea in three down at the break, but Mason Mount threw the Blues a lifeline with a goal 10 minutes into the second half.
Callum Hudson-Odoi made it 3-2 in the 70th minute and it took until stoppage time for Chelsea to find an equaliser when Abraham pounced on a loose ball in the box and slammed it home to snatch a point for Frank Lampard's side.
"We had a few chances in the first half, and they scored from every shot. We knew we had to come out in the second half and keep going, not drop our heads, and that's what we've done," Abraham told Sky Sports.
The 22-year-old striker described his side as a work in progress and said they would continue to improve.
"We need to cut out the mistakes as a team -- not just defensively... so we don't concede. We need to be strong together and keep encouraging one another," Abraham added.
Chelsea coach Lampard was disappointed his side left themselves with a mountain to climb.
"You can't legislate for pure mistakes, which they were today. Then it gives you a really difficult game to get back into," Lampard told Sky Sports.
"I can laud character, because we showed character and persistence, but we didn't need to if we did things properly in the game," he added.