Hillary Clinton took the lead in five key battleground states following the first presidential debate, giving her a solid boost with just over five weeks until Election Day, according to a poll released Thursday.

Voters in the five states — Colorado, Florida, North Carolina, Pennsylvania and Virginia — also declared Clinton the winner of the much-anticipated slugfest with Donald Trump, according to Public Policy Polling.

“Clinton has solid leads in Colorado, Pennsylvania, and Virginia — states seen as important to her path to 270 electoral votes — and modest leads in Florida and North Carolina, where wins would be indicative of a dominant overall victory in the Electoral College,” said pollster Tom Jensen.

“If these results hold up, Donald Trump has no path to victory.”

In four-way races that include the Libertarian and Green Party candidates, Clinton led Trump in Virginia and Colorado by identical margins of 46 to 40 percent.

She was also ahead in Pennsylvania 45 to 39 percent, Florida by 45 to 43 percent and North Carolina by 44 to 42 percent.

Analysts said it was clear that Clinton’s performance — and Trump’s underwhelming debut on the presidential debate stage — helped her numbers.

The poll “confirms the impression many people had of the debate and how it would play among undecided voters,” Julian Zelizer a history and public affairs professor at Princeton University, told The Post.

But Clinton’s camp shouldn’t be breaking out the champagne just yet, he added.

“Given how dramatically her lead narrowed last month there is clear evidence of a certain fragility with her margins. Also let’s remember if the first debate does have a real impact on the polls, that means the second one can as well. Democrats will remember how [Presidents] Reagan and Obama came back in 1984 and 2012 from spectacularly poor first debates,” Zelizer said.

Matt Hale, a political science professor at Seton Hall University, said the numbers give Clinton a stronger shot of winning.

“The national polls can tighten and yet be meaningless because what matters is the Electoral College and therefore swing state vote,” he said.

“Clinton has always had an easier path to 270 electoral votes than Trump. The recent swing state polls are really just confirming that easier path.”

Another poll showed that Trump’s performance damaged his standing with women voters.

The NBC News/SurveyMonkey poll showed that 27 percent of likely women voters said they thought worse of Trump after the debate.

Thirty percent said their opinion of Clinton had improved, while just 11 percent said they had a better opinion about Trump. Only 13 percent said their opinion of Clinton was worse.

“At the moment people, and especially women, were paying the most attention ever to the race Trump decides it is a good time to call a beautiful woman fat. It is a wonder he isn’t doing even worse with women,” Hale said, referring to the GOP nominee’s fat-shaming of ex-Miss Universe Alicia Machado.