Prior to the international break, reports emerged claiming that the Croatian had four games to save his job after the owners were disappointed with the club's start.
After the final whistle, Bilic told Sky Sports News: "I am happy. I am happy for the lads and for the club. We started with three away games and three defeats and this was a must win game.
"It was not a nice performance but it was a top performance in all other stuff like getting the ball, winning free-kicks and corners. We totally deserve it.
"I feel the pressure for a whole year now. It is maybe not logical for me to be under pressure after three games but it is modern football. We said to the guys we would go back to basics, we had to keep the clean sheet and stop them from having the ball in dangerous areas. We knew with three players up front they would always cause problems."
Next up for West Ham is a trip to The Hawthorns to face West Bromwich Albion.