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
9. Leo Trossard 5
Took up some decent positions but Brighton just couldn't get their most creative player in the game
10. Jeremy Sarmiento 5
His first Premier League start but it turned into a nightmare for the 19 year-old as he hobbled off with a hamstring problem after just 13 minutes. Gutting.
11. Neal Maupay 8
Had one half chance in the first that he dragged wide. Sublime pass to tee-up Moder for his big chance. Worked tirelessly and what a finish at the end
12. Solly March 6
On for Sarmiento and added control to the attacks. One deflected effort whistled just over the bar