The Magpies came from 2-0 down to secure a 2-2 draw on Saturday.
The Magpies looked set for a disappointing afternoon on Saturday as they found themselves two goals down to Liverpool at Anfield.
Instead, an impressive second half turnaround saw them bring the game back to 2-2 and secure a vital point from the jaws of defeat.
The result saw them go level with Sunderland, who play Arsenal tomorrow, and a point behind Norwich in 17th.
Hartson was impressed by the performance, saying they are not dead and buried just yet given the fight they showed on Merseyside.
"Newcastle have given themselves a fighting chance now,” he told BBC Sport.
“Credit to them - no matter what people say, on the evidence of this second-half performance they are not dead and buried yet."
Rafa Benitez will now be hoping that Arsenal can do them a favour in the upcoming games as they take on both of their rivals.
They take on Sunderland on Sunday, while next week sees them take on Norwich City, and should they win both of those games Newcastle United’s survival hopes would be given a huge boost.
First and foremost they must focus on winning their own games though, and given the spirit they’ve shown in the last two games, that seems more than a possibility.