Antonio opened the scoring inside the first 10 minutes for the Hammers and Caroll wrapped up all three points with a powerful header on 55 minutes.
Emre Can started for Liverpool and put in a typical hard-working performance. The 21-year-old completed the highest number of passes for his side (78) but at a disappointing 79 percent accuracy.
Can also contributed to attacking moves and was the main creative outlet pulling off six key passes and he even hit the post.
Bilic has a great record against Liverpool knocking the Reds out of the Europa League last season during his stint in charge of Beşiktaş last season.
The Black Eagles knocked Liverpool out of the Europa League Round of 32 on penalty shoot outs with Bilic in charge.
West Ham v Liverpool
Kick-Off: 12:45 (GMT)
Stadium: Boleyn Ground
West Ham: Adrian, Tomkins, Collins, Ogbonna, Cresswell, Kouyate, Lanzini, Noble (c), Antonio, Valencia, Carroll
Subs: Mignolet, Clyne, Lovren, Sakho, Moreno, Lucas, Can, Ibe, Firmino, Coutinho, Benteke
Liverpool: Mignolet, Clyne, Lovren, Sakho, Moreno, Lucas, Can, Firmino, Ibe, Coutinho, Benteke
Subs: Bogdan, Toure, Lallana, Allen, Brannagan, Smith, Randall
Goals: 10′ Antonio, 55′ Caroll