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.
Sealing the deasl: Newcastle's Jonas Gutierrez celebrates his goal
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.
Newcastle, still coming to terms with Andy Carroll's departure for Liverpool and without key men Cheick Tiote and Leon Best, created more than enough chances to win the game in the first half alone.
Karl Henry had spent much of the reverse fixture at Molineux in August reminding Joey Barton of his presence, and his team-mates took up the theme today.
Opening salvo: Kevin Nolan celebrates putting Newcastle in front
George Elokobi and Nenad Milijas were booked for early fouls on the midfielder and on both occasions the home side almost scored from the resulting free-kicks.
First Barton floated a fifth-minute ball into the box for defender Mike Williamson to send in a diving header which goalkeeper Wayne Hennessey had to palm around the post, and he then presented Nolan with a 16th-minute chance, but he directed his effort straight at Hennessey.
When the opening goal did arrive, it came straight down route one as Ameobi climbed well to flick-on Danny Simpson's long ball into the path of Nolan, who stabbed past Hennessey.
Wolves, who were busy in midfield but lacked any real creativity, were fitful in their response and it was not until Ebanks-Blake met Jamie O'Hara's 44th-minute free-kick in front of goal that Steve Harper was called upon to make a save.
Masked raider: Shola Ameobi (right) celebrates notching Newcastle's second goal
But Newcastle's response was emphatic and they doubled their lead in the final minute of the half when Ameobi climbed to power a header home from Lovenkrands' cross.
The only disappointment for Newcastle was the 44th-minute booking - his 10th of the season - collected by Nolan for a trip on winger Adam Hammill which will cost him a two-game ban.
Alan Pardew's men might have been home and dry within two minutes of the restart as they ran at will through a dangerously permeable Wolves defence.
First Hennessey had to make a vital block after Ameobi had forced his way to the goal-line and squared for Barton, and then the woodwork came to the rescue when Lovenkrands had rounded the keeper, but shot against the upright from a tight angle.
However, the game was over with 50 minutes gone when Fabricio Coloccini was allowed to run out of defence and find Barton down the right, whose low cross was tailor-made for Lovenkrands to help himself to his fifth goal of the season.
Three and easy: Peter Lovenkrands (left) adds a third goal for Newcastle
But Wolves refused to give up the fight and they grabbed a lifeline eight minutes later when Ebanks-Blake side-footed home from Jarvis' pinpoint cross.
Newcastle's composure eluded them as they attempted to deal with the setback, and they were fortunate not to concede again two minutes later when substitute Steven Fletcher glanced a free header wide.
For the first time in the game, the visitors had established a momentum, and substitute James Perch came to the home side's rescue twice within seconds, first dispossessing Ebanks-Blake inside the box and then clearing his goalbound header off the line.
The onslaught continued as the clock ran down, but Gutierrez relieved the tension when, after substitute Steven Taylor had blocked Karl Henry's shot and steamed upfield, he fed the midfielder to cut inside and curl an inch-perfect shot past Hennessey.
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Explore more:People: Kevin Nolan, Sylvan Ebanks-Blake, Adam Hammill, Karl Henry, Andy Carroll, Danny Simpson, Leon Best, Joey Barton, Jamie O'Hara, Steven Fletcher, Nenad Milijas, Wayne Hennessey, Steven Taylor, Jonas Gutierrez Places: Newcastle, Liverpool, Argentina