Following a 5-1 defeat to Chelsea in their last home game, the Hammers imploded after the break in this latest London derby.
West Ham are now winless in seven Premier League games at the London Stadium - their worst home run since 2003.
After a competitive and even first half, Spurs took charge when Pape Mata Sarr headed home from a corner - the fifth goal Potter's side have conceded from such scenarios this season.
Soucek was then given his marching orders for a high tackle which ripped the sock of Joao Palhinha, and the home side wilted.
Lucas Bergvall looped a fantastic header home to double the lead before Micky van de Ven swept home a third, who had to endure some Mohamed Kudus showboating in the closing stages as their former player relished his return following his £55m move to Spurs this summer.