The Hammers suffered an early setback when Wilfried Zaha put Palace in front after just three minutes.
But the stronger mentality that David Moyes has lauded in West Ham this season showed as Soucek struck twice in 16 minutes to turn the game around.
Only Manchester United's Bruno Fernandes has scored more Premier League goals from midfield than the Czech since he joined from Slavia Prague a year ago.
Soucek powered home a header from close range from Michail Antonio's cross and then bundled in Aaron Cresswell's free-kick at the back post.
"This is the performance that we wanted to show everyone and I hope that we can carry on," Soucek said. "Our dressing room is beautiful and we have so much confidence."
West Ham could have been out of sight by half-time as Antonio then twice struck the post.
A third goal was no more than the visitors deserved and it came 25 minutes from time when Craig Dawson headed in Jarrod Bowen's corner.
Michy Batshuayi pulled a goal back with virtually the last kick of the game but it was too little, too late for Palace.
A sixth consecutive win in all competitions takes Moyes's men above Liverpool in the table and Jurgen Klopp's struggling Reds will be wary of their trip to the London Stadium.
At the other end of the table, Newcastle's dreadful form continued as its winless run stretched to 11 games to pile more pressure on beleaguered boss Steve Bruce.
"This league is demanding and cruel at times," Bruce said. "I don't know if I've been on a worse run in 20 years, but tonight I've seen a little bit of something and we will start again tomorrow."
The Magpies have a seven-point cushion over the bottom three, but have now played two games more than Fulham in 18th.
Leeds had been on a poor run, but secured its first win of 2021 to move 14 points clear of the drop zone in 12th.
Raphinha slotted home Rodrigo's cut-back to give Marcelo Bielsa's men a deserved half-time lead.
Miguel Almiron levelled with Newcastle's first goal for five games early in the second half, but parity lasted only four minutes before Jack Harrison's sensational finish with the outside of his left foot claimed all three points for the visitor.