Bilic told BBC Sport after the game: "It was difficult, we knew it would be. We were missing our most quality players - not one, not two, but four out of six, if we're talking about the last third of the pitch.
"The guys did brilliantly. Bournemouth were better than us in the first 15 minutes of the second half but they didn't hurt us.
"We just deserved it. We were pushing and wanted it more. We're delighted to end the first game at this beautiful stadium with a win and a clean sheet."
West Ham's next game is against Astra Giurgiu in the Europa League on Thursday night.