The former Bolton man said that he felt the team worked well and rued missed chances from Calvert-Lewin and Tosun at the end of the match.
“When you finish the game strongly and put Liverpool under pressure then you consider yourself unlucky,” said Allardyce.
“Dominic’s chance at the end is the key one as that’s the best chance of the game for either side today and it went over the bar.
“The clean sheet and the defending gave us the basis to finish the game stronger.
“I thought we started ok in the first half but then we got a little bit apprehensive about who should we shut down.
“We started chasing the ball a bit too much and then when we got possession we started giving it away too easily.
“When our substitutes came on with fresh legs and a bit more determination because the other lads had run their legs off we started to get amongst Liverpool more.
“In the first half it was one hell of a save from their keeper off Bolasie and then two saves from Pickford.
“The second half for me was the pleasing thing.
“We dominated the last 15 to 20 minutes and created enough chances to win it.”