The Hammers have bounced back from their loss to Aston Villa on the opening weekend of the season, but it is successive defeats at the start of the new campaign for Palace, having also been overcome by Brentford in their division opener.
West Ham had the better of the early exchanges, with Michail Antonio passing up a good chance, before Mohammed Kudus came close with an effort in the 12th minute.
Palace were improving, though, and Eberechi Eze and Odsonne Edouard both had opportunities in the middle part of the opening first period, as the Eagles settled after a slow start.
Eze also rattled the crossbar late in the first period, but the two teams entered the half-time interval all square despite 17 attempts in the opening 45 minutes, with just one of those proving to be on target, demonstrating the issues in the final third of the field.
West Ham managed to score twice in quick succession in the second period, though, with Soucek firing his side ahead in the 67th minute, before Bowen doubled the advantage with an excellent finish five minutes later.
Palace continued to have bright moments in the final third, but the Eagles were unable to find the back of the net at Selhurst Park, and there will be concerns among the fanbase due to the start to the season and the number of key players who are being sold.
CRYSTAL PALACE VS. WEST HAM HIGHLIGHTS
67th min: Crystal Palace 0-1 West Ham (Tomas Soucek)
West Ham make the breakthrough in the 67th minute of the contest, with Soucek finding the back of the net from inside the box after a bit of untidy defending from Palace.
Aaron Wan-Bissaka did brilliantly down the right on his debut, and Bowen and Lucas Paqueta were also involved before Soucek made the telling contribution.
72nd min: Crystal Palace 0-2 West Ham (Jarrod Bowen)
West Ham double their advantage at Selhurst Park, as Bowen races towards the edge of the Palace box before finding the back of the net with an excellent finish.
Palace will be slightly disappointed with their defending there.
MAN OF THE MATCH - MAX KILMAN
Kilman's standout piece of work came in West Ham's second goal, with the centre-back providing a brilliant assist for Bowen, driving forward from the back before delivering a clever pass to his teammate down the right.
However, the centre-back's defensive work was also excellent, with Palace's forwards frustrated in a number of battles with the summer arrival.
CRYSTAL PALACE VS. WEST HAM MATCH STATS
Possession: Crystal Palace 58%-42% West Ham
Shots: Crystal Palace 14-18 West Ham
Shots on target: Crystal Palace 2-3 West Ham
Corners: Crystal Palace 3-3 West Ham
Fouls: Crystal Palace 9-16 West Ham
WHAT NEXT?
Palace will be in EFL Cup action on Tuesday evening at home to Norwich City, before resuming their league campaign away to Chelsea next Sunday.
West Ham, meanwhile, will tackle Bournemouth in the EFL Cup on Wednesday, before welcoming the champions Manchester City to the London Stadium in the league next Saturday.