The Black Cats scored two goals in the final 10 minutes after trailing 2-0.
"The final-third quality that has been missing was there - if we get that then we win games," Allardyce said.
"We created far more chances against Manchester City than we did [against Liverpool] but today we showed the kind of finishing I was looking for."
The draw meant Sunderland have picked up eight points from their last six Premier League games, compared to just three points from the previous six.
Allardyce's side remain 19th in the table, four points from safety and with the worst defensive record in the league.
But the ex-West Ham manager remains optimistic after bolstering his squad in the January transfer window.
"Never ever say die. You've got to go until the end," he added. "I'm really pleased about the new guys coming in from abroad into the side,
"Hopefully they're going to drag the rest of the players along with them with their bit of extra quality."
Manager Sam Allardyce says Sunderland showed the quality they will need in their relegation fight as they claimed a crucial point against Liverpool.