The two sides met at the Boleyn Ground on Wednesday evening in their FA Cup quarter-final replay, after the 1-1 draw at Old Trafford.
It was Louis van Gaal’s side that progressed, with goals from Marcus Rashford and Marouane Fellaini seeing his side progress to meet Everton in the semi-finals.
It was a disappointing end for the Hammers, who had been eyeing the trophy as the perfect send-off for their famous old ground.
And Bilic has conceded that the away side were simply better than his team, but has vowed they will bounce back as they hunt down a top four place.
"It is not what we wanted of course, but we gave our best,” he told BBC Sport.
“They were better tonight in the majority of the game. They were the better side and we have to admit it.
"We are fighting to finish as high as possible in the league, we have seven big games coming. It takes out a bit of energy and motivation but we still have a big thing to play for. We are going to bounce back."
The Hammers must now turn their attention towards their attempts to secure a top four place this season.
They currently sit sixth in the table and give points behind Manchester City, who currently occupy that crucial fourth place.
With many points left to play for however, that Champions League place is very much up for grabs, and they will be keen to make it their own after their disappointing exit from the FA Cup.