The hosts were two goals to the good after just six breathless minutes thanks to Winston Reid and Diafra Sakho, and added a third through Morgan Amalfitano late on after Raheem Sterling had given Liverpool hope.
Allardyce praised the tireless work rate of his players following the win, hailing their efforts in the opening moments as crucial in setting the tone.
"Our higher pressing game was the reason we dominated Liverpool so much in the first half and to score two goals in that early period really stunned the Liverpool players," he said.
"We carried that right through to the end of the first half and we really took them to task. We played outstandingly well and didn't let Liverpool play at all so the only disappointment was that it was 2-1 at half-time and not more.
"Having said that, we've won the game, we deserved to win it.
"We've had to give a more solid defensive performance in the second half when Liverpool were going to throw caution to the wind and we've managed to pop up with the third goal at just the right time.
"Outstanding performance today, particularly the first half, and one where we've deserved everything we've got."
Allardyce went on praise the contribution of his summer signings to a morale-boosting win over title-chasing opposition.
Goalscorers Sakho and Amalfitano were among those brought to the club during the recent transfer window, as were starters Alex Song, Enner Valencia and Cheikhou Kouyate.
"It's [Alex Song's] first competitive game that he's started today. He didn't play any games in pre-season for Barcelona, so I thought he did tremendously well with the pace of the game to last as long as he did," Allardyce added.
"The front two looked like they'd been here for a long time. The energy that they've got, not just in possession but out of possession I think terrorised Liverpool's back four.
"And we've got Cheikhou Koyate who loves football, loves life, loves being in every day. He's looking like a promising player for us. We've got really good players that we've recruited this season and that has lifted everyone else for me.
"That performance today just shows what level we might be able to achieve this season."
