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Benitez was unveiled as the club's new manager on Friday, hours after Steve McClaren was sacked.
"I don't think they would have stayed up if it had remained the same, but Benitez has given them a big opportunity," Shearer said.
Newcastle are 19th, one point from safety, but have a game in hand.
Shearer, who is Newcastle's record goalscorer and also managed the club in an ill-fated spell at the end of the 2008-09 season, said Benitez must be allowed to have full control over the team.
"It's a great coup for Newcastle to get someone like Benitez, if - and it's a big if - he can manage the football club the way he wants to manage," Shearer said.
"What's been clear is that every manager that Newcastle have appointed has come from a point of weakness, where they've not been able to set their own agenda.
"They've always had to work to a certain structure, whereas I think Rafa will be different - he'll set what he wants, he can demand that he controls the transfers, and he can run the football club as he wants to run it."
Benitez has previously managed Liverpool and Chelsea, winning the FA Cup and the Champions League with the former and the Europa League with the latter, and Shearer believes that record of success will endear him to Magpies fans.
"When you look at his success rate, almost wherever he's been he's delivered trophies," Shearer said.
"[Newcastle fans] will love anyone who gives them success. He's shown he can manage in tough situations.
"They're crying out for something to grab hold of up here. They just want people to sweat blood for the black and white shirt."
However, Shearer criticised the treatment of McClaren, whose sacking was widely reported in the media before it was confirmed on Friday.
"They did it all wrong," Shearer said. "The way they've treated McClaren this week has been so disrespectful.
"If they'd decided to make that decision, which obviously they had, they should have done it last Saturday night after the game, or Sunday morning.
"It was so unprofessional and so hard for Steve to take, and he shouldn't have been put through that."
You can listen to the full interview with Alan Shearer on BBC Radio 5 live shortly after 19:00 GMT.
The appointment of Rafael Benitez is a "great coup" which gives Newcastle a chance of staying in the Premier League, says club legend Alan Shearer.