Roberto Firmino headed the Reds in front after James Milner’s clever cross before the goalscorer turned provider to allow Adam Lallana to tap home.
Fans were pouring out when Reds goalkeeper Simon Mignolet was beaten by Adam Johnson’s curling free-kick.
Jermain Defoe completed the comeback after finishing from 10 yards.
Sunderland remain four points from safety and have now gone 11 Premier League games without a clean sheet, longer than any other side currently.
The Black Cats still have to host Manchester United, Leicester City, Arsenal and Chelsea in addition to facing relegation rivals Newcastle and Norwich away.
However, Sunderland overcame the early loss of striker Duncan Watmore, who hobbled off injured, to claim a point that did not seem likely after Lallana’s 70th-minute goal put Liverpool 2-0 in front.
Allardyce spoke in his post match press conference and said: “You think your day is over when you are 0-2 down at Anfield. But we fought back with two quality goals and got a point by not giving up. It was never say die until the end.”
The result came as a precious point for the boys in red and white.