Brighton were 1-0 behind and down to 10-men after injuries to Jeremy Sarmiento, Adam Webster and Adam Lallana.
Maupay, who was heavily criticised after the 0-0 draw with Leeds United for missing chances, produced a remarkable instinctive finish as he fired home an overhead kick from substitute Tariq Lamptey's cross
Soucek had earlier netted after just five minutes as the Czech international rose highest in front of Albion goalkeeper Robert Sanchez and glanced home his header from a dangerous inswinging corner.
Albion dominated possession and fashioned two decent chances in the first half. Maupay's was the more difficult as the Frenchman swivelled and dragged his effort just wide.
Jakub Moder also had a shot saved from Fabianski when the Poland international was clean through on goal and the experienced Lallana wasted a glorious opportunity on 67 minutes as he shot wide.
Scroll down and click through to see who shone and who struggled for Albion at the London Stadium
1. Robert Sanchez 5
The Spain international was beaten at the near post from the corner and distribution was shaky in the first half. Had a VAR let off when he flapped at another corner in the second half. Good save from Decland Rice on 81 minutes
2. Joel Veltman 5
A very Veltman-like performance. Solid and reliable with out being spectacular. Decent with the ball. Replaced by Lamptey with 20 to go
3. Adam Webster 5
Pulled up with a calf injury and replaced by Duffy on 34 minutes. Bit of a nightmare for Webster who had just returned from a hamstring problem
4. Lewis Dunk 6
Defended bravely against a physical and dangerous West Ham attack. Tried to get Albion moving forward by playing out from the back