The two teams drew 1-1 at Old Trafford on Sunday after Dimitri Payet's stunning free-kick was cancelled out by Anthony Martial's controversial equaliser late on.
While reluctant to "moan" about referee Martin Atkinson, Bilic felt Payet should have been awarded a penalty after being brought down by Marcos Rojo prior to his free-kick and believes goalkeeper Darren Randolph was fouled by Bastian Schweinsteiger during United's leveller.
Asked if West Ham should have had a penalty, he replied: "Yes. I'm not moaning now and I don't want to spoil the game by moaning about the referee.
"There were two crucial decisions. Firstly, the penalty on Payet. For me it is a penalty, Rojo slides from behind and is nowhere near the ball - a clear penalty.
"The second one is an obvious foul from Schweinsteiger on Randy - he would have definitely got to the ball."
READ MORE | Martial saves the day... and Van Gaal's job
Defeat for holders Arsenal earlier on Sunday has the competition wide open - with Crystal Palace, Everton and Watford the confirmed semi-finalists - and Bilic is optimistic that his side will join them.
"We played a really good game and I'm a little bit gutted because of the time we conceded the goal," he told BBC Sport. "We could have held on for 1-0 and on counter-attacks I was expecting to create one, two or three more clear-cut chances.
"Our dressing room was, and still is, very silent. I'm not over the moon, but it means we are moving in a good direction.
"Here, it is very hard to play with 75,000 fans and this atmosphere. Manchester United are a different team when they play away and don't have that confidence.
"It [the replay] is going to be a tough game, but I fancy us because we are going to be even more confident.
"With the history of our stadium it's going to be another historic game and you can't ask for more, but to make history we have to beat them - we won't put it in the books if we lose the last game.
"It's going to be another epic home FA Cup match and we couldn't ask for more, it's like Spielberg!"