West Ham travel to fellow high-flyers Leicester City in the Capital One Cup third round on Tuesday after securing away wins at Arsenal, Liverpool and Manchester City in the league.
That remarkable sequence - along with a home victory against Newcastle United - has them third in the standings after six matches, and Bilic is relishing a return to knockout competition.
West Ham's last major honour came when they lifted the League Cup in 1981 and, although Bilic acknowledges that his team are not among the favourites, it is a feat he is keen to emulate.
"We have the game on Tuesday and it is a big game for us. It is a competition where we know we have a chance," the former Croatia boss told a pre-match press conference.
"You never know what will happen but what I can say is that we will give our best. We are approaching the competition with the aim to win it if possible. We are not the favourites but that will be our approach."
Bilic confirmed that he will rotate the line-up that triumphed against Man City at the Etihad Stadium, although the cup-tied status of Nikica Jelavic and Michail Antonio is a source of frustration.
He added: "We will try to look at other players. It is a shame that Michail and Jelavic cannot play as they have already played in the competition and are cup-tied.
"But we will try to maintain our chances and keep the same quality on the pitch.
"If possible it will be good to rest a couple of players who played against Newcastle and Manchester City on Saturday."