The Hammers had pressure on them to pick up all three points after learning that Manchester United took another step closer to Europe with a 1-0 win over Norwich City at Carrow Road.
The East London club headed in to the game in fine form, and they dominated the early exchanges.
However, two goals in quick succession from Wayne Routledge and Andre Ayew shocked the hosts and things only got worse the more they tried to push forward in an attempt to get back into the match.
Further goals from Ki Sung-Yong and Bafetimbi Gomis either side of a Stephen KIngsley own goal gave Swansea a resounding win, leaving Bilic’s men in sixth place, five points off the top four.
"I keep saying the gap between us and the top four is bigger than the gap between us and Southampton and Liverpool below. That's the gap that worries me and that's the gap I'm still concentrating on,” Bilic told the London Evening Standard.
"But we have a game on Tuesday and we are still in a brilliant position. We want to finish high. With two games to go we are talking about Europe. It's in our hands. It's there for us."
Fortunately for West Ham, they still have a Europa League spot in their hands and they can quickly put the Swansea defeat behind them with a midweek home encounter with Manchester United.
Four points separates the Hammers and the Red Devils, and victory for Bilic’s men will give them the chance to leapfrog United next weekend when they face Stoke City at the Britannia Stadium.