The 28-year-old takes over from Liverpool’s Steven Gerrard, who retired after England’s World Cup elimination.
Rooney has 40 goals in 95 appearances, and was named club captain by United boss, Louis van Gaal, earlier this month.
“It is something I will take massive pride in doing. To be appointed captain is beyond my wildest dreams,” said Rooney on his official website.
England play Norway at Wembley on September 3 in a friendly before their Euro 2016 qualifying campaign starts in Switzerland on September 8.
Hodgson has named four uncapped players in his latest squad, with Arsenal’s 19-year-old defender, Calum Chambers, and Newcastle’s Jack Colback, Tottenham’s Danny Rose and Aston Villa’s Fabian Delph, all 24, in line to make their international debuts.
Speaking on Rooney’s appointment, Hodgson said: “I gave it a lot of thought but Wayne is an obvious choice.
“He’s the right man and he’ll show enormous enthusiasm and responsibility.”
Rooney became the youngest player to play for England when he made his debut in a friendly against Australia on 12 February 2003, and scored his first goal later that year against Macedonia in a European Championships qualifier.
He has captained England twice – in a pre-tournament friendly with Brazil in 2009 and a World Cup qualifier against San Marino in 2012 – as well as having previously had a stint as vice-captain.
Rooney added: “As a kid I always loved the occasion of watching a big England game on the television. Back then I had a burning ambition to play for my country. Now to be appointed captain is beyond my wildest dreams.
“I would like to thank Roy and the coaching staff for having the belief and confidence in me to do the job.”
Rooney succeeds 34-year-old Gerrard, who revealed he had been left “broken” by the failure of a World Cup campaign in Brazil.