The Reds, who were in European action on Wednesday night, started slowly at Upton Park with the hosts racing into a two-goal lead within the first seven minutes.
"When you see a game plan work as early as that, it's brilliant," Allardyce told BBC Sport. "Liverpool had played in the Champions League for the first time in five years which is tiring and we wanted to start as quickly as we can.
"We said let's not let them control the tempo and press their back players. It worked brilliantly. From there on they only had one shot on target and we could have had three or four.
"It's a fantastic result and performance. I don't think anybody across the world of football can say that we didn't deserve that today."
West Ham return to action next weekend when they travel to face Manchester United at Old Trafford.