Chelsea started brightly and took the lead thanks to Christian Pulisic’s close-range finish in the 27th minute after Marcos Alonso’s initial free-kick struck the post.
But Thiago Silva was given his marching orders for a second bookable offence just moments later and Matheus Pereira and Callum Robinson both completed braces for the visiting Baggies at Stamford Bridge.
Mason Mount and Mbaye Diagne were the other names on the scoresheet on an afternoon to forget from Chelsea’s perspective and the result will give London rivals West Brom and Tottenham hope of catching them in the race for a top-four finish.
Meanwhile, West Brom, sitting 19th in the table, moved to within seven points of 17th-placed Newcastle after the stunning win and Allardyce was smiling from ear to ear as he spoke to BT Sport shortly after the final whistle.
‘I think we’re really pleased and satisfied – or I am – of how the players applied our tactics today, how we set out to control Chelsea’s attacks and then of course what we could do in possession of the ball,’ he said.
‘I have to say because we ended up being so good in possession of the ball on so many occasions and our quality of finishing at the end of today, which was the best it’s been since I’ve been here, it was a thoroughly deserved victory.
‘People will always look to the fact that the went down to 10 men and so on and so forth but I’ve been manager of many teams that have won and not lost so we took full advantage after that.
‘But I thought we started the game really brightly and I thought we were creating chances.’
The Baggies head coach added: ‘I think Silva was lucky to stay on when he pulled Pereira back with both of his hands, going through maybe we would have scored a goal.
‘And then we had to recover from a goal they scored and we did that brilliantly today.’
Asked how he rates West Brom’s survival chances, Allardyce replied: ‘A little better after three points but still a long way to go for us.
‘Six out of nine I said and now it’s five out of eight and we might have a chance. Even then we might need a couple of draws or a draw.
‘But we’ve started on the track of trying to get an undefeated run, trying to get more wins in the last eight games than we’ve had all season.
‘It’s been done before, this is a good start but there’s a long way to go still.’
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