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Pope, 30, has reportedly cost Newcastle around £10million and is expected to replace Martin Dubravka as Eddie Howe's number one.
The former Charlton Athletic player has signed a four-year contract at Newcastle and becomes the Magpies' second confirmed deal of the close season, after Matt Targett's loan move last term was made permanent.
Pope, capped eight times by England, continued to impress at Burnley even as the Clarets headed for the Championship. It was inevitable he would leave after they went down, given his desire to play at the World Cup later this year.
Among goalkeepers with at least 10 appearances in the 2021-22 Premier League, Pope had the fourth best save percentage (71.69 per cent).
He managed nine clean sheets from 36 appearances, which was more shut-outs than Manchester United's David de Gea (eight from 38 games) and West Ham's Lukasz Fabianski (eight from 37 games).
While at Burnley, where he spent six seasons, he was named in the 2019-20 Premier League team of the year after keeping 15 clean sheets, which was bettered only by Manchester City's Ederson.
Head coach Howe said: "Nick is an exceptional Premier League and international level goalkeeper, so I'm very pleased to be adding strong competition to a very important position.
"There has been a lot of interest in him from other clubs this summer, so I'd like to thank our owners and everyone behind the scenes for getting us ahead of the competition.
"I'm delighted he is joining us as we prepare for the exciting challenge ahead."
Pope is relishing the challenge, with Newcastle having enjoyed a revival in the second half of the season following an influx of transfer funds from the club's new Saudi owners.
From looking like relegation fodder, Newcastle raced up the table to finish in 11th place, while Burnley ended up 18th in the 20-team table.
Pope said Newcastle had been "taking good steps in the right direction in the recent past".
"This club maybe underperformed in the past a little and would like to perform better. The ambitions are there to go to the next level," Pope added in an interview on the club's website.
"It's something for myself that's really exciting to be a part of."
Newcastle confirmed the long-anticipated arrival of sporting director Dan Ashworth in May, after he left Brighton and Hove Albion, and have been strongly linked in recent weeks with Lille defender Sven Botman and Reims forward Hugo Ekitike.