West Ham grew into the match after a sluggish start at The Hawthorns and Cheikhou Kouyate's header coupled with Mark Noble's brace - his second in as many Premier League matches - which helped them to a comfortable three points.
The Hammers have now surpassed their record points total in the Premier League, while a place in Europe's secondary competition remains a realistic possibility as they sit above Manchester United on goal difference in fifth.
West Ham face Louis van Gaal's men in between fixtures against Swansea City and Stoke City, and manager Bilic is targeting a strong finish.
"There are still nine points to play for and we are not big headed, but we have to say now that the realistic target for us is a European spot," he said.
"It's going to be hard because there are still a few teams who want the same and the gap is not big. All we can do is prepare ourselves for the next game and try to beat Swansea."
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Kouyate and Noble may be two of West Ham's less likely goalscorers, but Bilic believes that is proof of how well his side performed as a team.
"When you have two of your holding midfielders scoring the three goals it shows you are playing good football," he added.
"They weren't set play goals either, they were there because we kept the composure, we kept the ball until they came to the box. The opponents were more and more occupied with Andy [Carroll] or [Diafra] Sakho, so we gave them a little more of a gap for them to come."
Bilic was late turning up to his news conference as he was catching up with opposite number Tony Pulis, who he feels is among the best coaches in the Premier League.
"I was on my way to the press conference and then Tony stopped me and I had no option!" he said. "I love him. When I was in Turkey we had great conversations about everything and I rate him the highest in England. We had a great chat. He is brilliant."