The Premier League clash at Goodison Park finished goalless, but the Toffees spurned four gilt-edged opportunities in the latter stages of the game.
"I think that in the end when you finish the game as strongly as that and put Liverpool under pressure, you can say we're unlucky, particularly the Cenk Tosun header and Dominic Calvert-Lewin's chance," Allardyce told Sky Sports News.
"The basis of our performance was nullifying Liverpool's attack to create chances but when you miss those chances, it really is disappointing because if they had gone the other way, we could've been looking at a different game.
"I think we got apprehensive about chasing the ball too much and then giving it away too easy. In the second half, we played much better, we took it to Liverpool and ran at them to create chance after chance."
Everton remain ninth in the table, three points clear of 10th-placed Bournemouth, following the stalemate.