The visitors left themselves with a mountain to climb after Kevin Nolan, Shola Ameobi and Peter Lovenkrands put the Magpies 3-0 ahead with just 50 minutes gone.
They eventually staged a fightback as Sylvan Ebanks-Blake reduced the deficit with 58 minutes gone and then saw a 72nd-minute header cleared off the line.
But Argentina international Gutierrez made sure with a superb solo effort in injury time to the delight of a crowd of 49,939 to take the Magpies' points tally to 39 and leave them within touching distance of safety.
It was just the Magpies' second win in 10 league games and brought an end to Wolves four-match unbeaten run.
Source: PA
Source: PA