"My feeling is satisfaction," he told his post-match news conference.
"The game was even in the first half. Liverpool were the better side in the second half until the final 10 or 15 minutes. Our keeper made a couple of saves and so I was satisfied."
Goalkeeper Simon Mignolet was the only Liverpool player to retain his place form Tuesday's League Cup semi-final triumph over Stoke City, with Klopp opting to push some of his more youthful squad members to the fore.
The likes of Cameron Brannagan, Kevin Stewart and Brad Smith turned in eye-catching performances, leaving Bilic content to blunt a Liverpool attack for the third time this season.
"Maybe people were not expecting this Liverpool team to play that well or match us but, on the other hand, Liverpool when they play at home in their shirt are a big team so I am happy," he added.
"It is always hard to get a clean sheet. We didn't want a replay but we can live with that schedule. One more game isn't going to be a big problem for us.
"We deserve a replay and for that reason I have to congratulate my players. It will be a great night to play Liverpool at Upton Park. It will be one of the historical games."
Bilic lost centre-back James Tomkins and midfielder Chekihou Kouyate to first-half injuries.
He expects Kouyate to recover from a minor problem to face Aston Villa in the Premier League on Tuesday, although Tomkins required stitches in head injury and is set to miss out.
"Tomkins had 13 stitches," Bilic continued. "Kouyate then came off but he should be okay for Tuesday.
"When you have to make two changes early on it affects your plans. Tomkins has stitches and will be doubtful for Tuesday."