Paxman bowed out of the BBC2 news programme after 25 years in June.
BBC director general Tony Hall said that Davis, best known for his role on Dragons' Den and Radio 4 flagship breakfast news show Today, would do an "outstanding" job.
The news comes after a turbulent period for Newsnight in which an investigation into Jimmy Savile's sex crimes was dropped and a separate story led to Lord McAlpine being wrongly accused of child abuse.
The show's female presenters, Laura Kuenssberg, Kirsty Wark and Emily Maitlis, had all been tipped for the plum role.
The director general said: "Evan is an outstanding journalist, an extraordinarily clever and intelligent interviewer.
"He has a wonderful presence on TV. I've got no doubt he will be a really great presence on Newsnight."
Davis said on Twitter: "I didn't believe it when I read about it in the papers but now it's official. I'll be leaving @BBCr4today and heading off to Newsnight.
"I had been determined to outlive John Humphrys on @BBCr4today so I have failed. But you can't turn down Newsnight."
Davis will be the anchor for the programme three nights a week from the autumn.
A former economics editor on the programme, he has been a Today programme host for six years. Kirsty Wark, Emily Maitlis and Laura Kuenssberg will all continue to present Newsnight.
The programme's editor Ian Katz said: "Evan brings a rare combination of curiosity, intelligence and mischief to his broadcasting - just the qualities I want Newsnight to be known for. I've admired him from afar as a listener and viewer for years, and I'm thrilled that he's joining us to help reinvent the show where he made his name in the 90s."
Davis - formerly an economist before joining the BBC - said of his new roles: "While it is a scary prospect, it will be an adventure and a challenge, and I hope the viewers will be happy with the result."
He pointed out that he felt "terribly sad" to leave his present role and he paid tribute to the team: "I have been there for over six years and those have been some of the most satisfying of my career.
"That programme is put together by remarkably few people, and I can't praise them highly enough for their professionalism and companionship," added the Oxford graduate, who also hosts BBC2's Dragon's Den.