The Baggies headed to the capital with their hopes of staying in the Premier League having appeared to have gone after they failed to beat Newcastle or Crystal Palace in their last two.
Chelsea meanwhile were full of confidence with the Blues unbeaten and having conceded just two goals in 14 games since Thomas Tuchel replaced the sacked Frank Lampard as boss.
It felt as though the hosts would go on to secure a routine victory when Christian Pulisic put them ahead just before the half-hour mark.
But Thiago Silva was sent off for a second bookable offence just a few minutes later.
And Albion really made them pay.
In first-half stoppage time, Matheus Pereira scored two outstanding goals to put them in front.
Strikes from Callum Robinson and Mbaye Diagne saw Sam Allardyce's side race into a 4-1 lead after the interval.
Substitute Mason Mount then pulled a goal back for the Blues.
But in stoppage time, Albion made the scoreline emphatic with Robinson producing another exquisite finish to make it five.
The result means with eight games to play, Albion now sit seven points from safety.
Allardyce sprung a surprise when naming his starting line-up with the boss opting to move away from the 4-3-3 system that had made Albion much more defensively sound in recent outings.
With e eye on Chelsea's innovative formation, the 66-year-old decided to switch to a back three with Semi Ajayi lining up alongside Dara O'Shea and Kyle Bartley at the heart of the defence.
In what, overall, was a 3-5-1-1 system, Darnell Furlong and Conor Townsend operated as wing-backs with Pereira just behind Diagne up front.
In central midfield, Matt Phillips lined up alongside Okay Yokuslu and Ainsley Maitland-Niles - with loanee Conor Gallagher unable to feature against his parent club.
Chelsea lined up in a 3-4-2-1 system with Pulisic and Hakim Ziyech both deployed in the number 10 role.
Timo Werner was the lone centre-forward with Reece James and Marcos Alonso the widemen.
In what proved to be an incident-filled first half, it was the Baggies who started on the front foot with Pereira sending a 20-yard strike straight into the arms of Edouard Mendy after just two minutes.
The hosts were intent on playing out from the back but they survived a huge scare when Jorginho passed the ball straight to Pereira.
The Brazilian darted at goal before being hauled down on the edge of the box by Silva - who escaped with a yellow card despite a VAR check to see if it should have been a red.
Gradually Chelsea began to grow into the game with Marcus Alonso seeing a low drive claimed by Sam Johnstone.
But while Tuchel's side looked threatening in possession, Albion had arguably been the better team as the half reached its midway point.
Suddenly, though, things started going against them.
O'Shea went down injured and had to be replaced by Branislav Ivanovic.
Chelsea then took the lead when Alonso smashed a free-kick against the post but Pulisic reacted quickest and turn the ball home.
Albion went straight up the other end with Silva once again producing a reckless challenge on the edge of his own box - this time on Yokuslu.
The Brazilian had been a touch fortunate not to have been sent off earlier, but now he had left referee David Coote no choice with the defender given his second yellow card of the game.
Albion lost Ivanovic just a few moments later - the centre-back having to come off just 12 minutes after coming on due to pulling a hamstring following a foot race with Werner.
But with the hosts down to 10-men, Allardyce was able to be braver with his substiution.
Callum Robinson replaced the Serbian in a move that saw the Baggies switch to a 4-2-3-1 system.
And in first-half stoppage time, Pereira scored two outstanding goals ot put Albion ahead.
He pulled the Baggies level after collecting a long punt forward from Johnstone before brilliantly lifting the ball over Mendy and into the net.
And then with just seconds remaining, the Brazilian brilliantly stroked the ball into the bottom corner from 20-yards to completely turn the game on its head.
Chelsea made a change at the break with Mason Mount replacing Pulisic.
The hosts then came a whisker away from pulling level when Johnstone failed to claim a cross from James.
The ball fell kindly for Alonso whose powerful strike was cleared by Ajayi on the goal-line.
Albion rallied though and should have made it 3-1 when Pereira entered the box and really should have picked out Phillips but instead saw a low drive saved by the legs of Mendy.
Just minutes later, though, the Baggies did have a two-goal cushion thanks to yet another outstanding finish.
Furlong whipped in a cross from the right which Robinson met to spectacularly volley the ball home from 12-yards.
Albion were now in dreamland with Diagne making it 4-1 in the 68th minute.
After breaking swiftly from a Chelsea corner, Townsend found Pereira in the box who then moved the ball on to the Senegal international.
And from 18-yards he made no mistake with his powerful drive flying past Mendy and into the net.
To Chelsea's credit, they have themselves some hope just three minutes later when Mount tapped in a cross from Werner.
Chelsea tried to pile on the pressure with 20 minutes still to play.
Albion dealt with them fairly comfortably though.
And in added-on time they grabbed a fifth when Pereira claimed his second assist with a wonderful through-ball.
Robinson latched onto it and exquisitely dinked the ball over Mendy for his second of the game and fourth against Chelsea this season.
Chelsea (3-4-2-1): Mendy, Azpilicueta, da Silva, Zouma, James, Jorginho (Havertz 72), Kovacic, Alonso, Ziyech (Christensen 33), Pulisic (Mount 45), Werner.
Subs not used: Arrizabalaga, Rudiger, Hudson-Odoi, Chilwell, Gilmour, Giroud.
Albion (3-5-1-1): Johnstone, Furlong, O'Shea (Ivanovic 24, Robinson 37), Ajayi, Bartley, Townsend, Phillips (Livermore 89), Yokuslu, Maitland-Niles, Pereira, Diagne.
Subs not used: Button, Peltier, Diangana, Snodgrass, Grant, Robson-Kanu.