Colin Graves, chairman of the England and Wales Cricket Board (ECB), said in a statement that the Yorkshire batsman had been the perfect choice.
"I’m delighted that he has accepted the role and will now help to take the team to the next level," he added.
Durham's 25-year-old all-rounder Ben Stokes was named vice-captain, the role previously held by Root.
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Root, 26, has scored 4,594 test runs since his debut in 2012 and will become England's 80th test skipper with his first game in charge set to be against South Africa at Lord's on July 6.
Cook, 32, resigned on Feb. 6 after captaining England for a record 59 tests.
Root, third in the world test batting rankings, had been the standout candidate despite concerns that the extra responsibility could affect his form.
The Yorkshireman described the new role as a "huge honour".
"We have a very good group of players and I’m looking forward to leading them out in the summer, building on Alastair’s achievements and making the most of our talents in the years ahead," he said.
"The senior guys in the changing room play a very influential role and whilst there’s a natural progression for me it’s a huge support to know that they are there to help and advise."
Root made his test debut in December, 2012 at the age of 21 and has scored 11 centuries in the longest format of the game.
The ECB said director of England Cricket Andrew Strauss and head coach Trevor Bayliss had held conversations with three players before the selectors made their decision.
Root did not hesitate when the offer was made.
"Joe is the right man to be our next test captain and I’m thrilled that he has accepted the role," said Strauss.
"He is universally respected by his team mates, passionate about driving the test team forward and extremely excited about the prospect of leading his country.
"I’m also delighted that Ben Stokes will take over from Joe as vice-captain of the team. He has real presence and influence within the team environment that serve as a great source of support for Joe."
England face a busy year with a home series against South Africa and West Indies followed by the Ashes at the end of 2017 in Australia.
(Reporting by Alan Baldwin; Editing by Ed Osmond)
