Callum Robinson's arrowed finish gave West Brom a fourth-minute lead and the striker capitalised on a loose touch from Chelsea debutant and captain Thiago Silva to double the advantage on 25 minutes.
West Brom had a third two minutes later as Kyle Bartley converted Darnell Furlong's flick at a corner.
Mason Mount's long-range shot gave Chelsea hope 10 minutes after half-time. Callum Hudson-Odoi added a second after a neat exchange with Kai Havertz.
And three minutes into time added on, Abraham tapped in Sam Johnstone's parry as Chelsea came from 3-0 behind to avoid defeat for the first time in the Premier League.
A first draw of the 2020/21 season moves Chelsea to sixth with four points. West Brom's first point moves them out of the bottom three.
King of the Match Mason Mount
West Brom: 4 Oct v Southampton (A)
Chelsea: 3 Oct v Crystal Palace (H)