The Hammers put three past their opponents without reply at The Hawthorns to move into fifth in the Premier League, although Manchester United can climb back up one place with a win against Leicester City on Sunday.
Liverpool and Southampton are also still in the race for a Europa League berth, but Bilic knows that nailing down the position is now in West Ham's hands as they host the Red Devils before the season is out.
"Now a European spot is a realistic target for us," he is quoted as saying by ESPN. "It wasn't at the beginning of the season but we're not going to deny it or be big-headed.
"Unfortunately it's not safe still and the gap between us and Liverpool and Southampton is less than the gap to the Champions League. We are talking about Europe, it's all in our hands and it's very simple. You don't have to be a genius to do it.
"If we beat Swansea City our target is going to be closer for us and then we have a game against Manchester United. That is and was our approach from the start of the season and it's the only logical approach. There's no point to calculate but we are in good form."
As well as facing Man United and Swansea, West Ham - who crashed out of the Europa League last year at the qualification stage - must also travel to Stoke City on the final day.