The striker clinched a 3-3 draw after Frank Lampard's side looked beaten at the break.
Mason Mount and Callum Hudson-Odoi struck following the restart to help break Baggies hearts.
Kyle Bartley was left unmarked from a corner to score a third as Chelsea were exposed.
Timo Werner hit the bar when they only trailed 1-0 and, while it was another embarrassing display at the back, Chelsea can take solace in their comeback.
Albion missed out on their first Premier League win of the season having lost heavily to Leicester and Everton.
Willy Caballero replaced the dropped Kepa Arrizabalaga but Chelsea's defensive problems continued as they fell behind after just four minutes.
Alonso cheaply lost possession to Matheus Pereira and he burst forward to find Robinson who drilled in a fine low drive into the corner.
Lampard has overseen just one away clean sheet in 21 Premier League games and will hope new goalkeeper Edouard Mendy, after his arrival from Rennes, will help to solve their problems.
Sam Johnstone parried Kai Havertz's 20-yard effort and Abraham fired wide from close range as the visitors tried to respond.
Abraham should have done better and Werner went closer after 15 minutes when he clipped the bar from Mount's cross.
But any hopes of an instant recovery were ended by more defensive howlers as the Baggies scored twice in three minutes.
First, under no pressure, Silva allowed the ball to run under his foot and Robinson went through to finish past Caballero.
It quickly got worse for the visitors after 28 minutes when Darnell Furlong headed Pereira's corner to the unmarked Bartley to volley in from close range.
Even with their attacking riches there seemed no way back for Chelsea and Grady Diangana had the chance to make it 4-0 just before half-time but shot over from 18 yards.
Lampard replaced Mateo Kovacic and Alonso for Hudson-Odoi and Cesar Azpilicueta at the break.
Chelsea emerged with greater urgency and they began their comeback 10 minutes after the restart when Mount spun away 25 yards out and fired into the bottom corner past a static Johnstone.
The goalkeeper turned Havertz's deflected effort wide immediately afterwards and resilient Albion tried to hold firm until Hudson-Odoi made it 3-2 with 20 minutes left.
The substitute swapped passes with Havertz to send him into the area and he confidently stroked across Johnstone.
Romaine Sawyers shot wide for Albion but they could not survive another late onslaught and Abraham rescued Chelsea in injury time when he tapped in after Johnstone parried Mount's shot.