The 30-year-old striker, who also captains Manchester United, has been England's skipper for the past two years.
He will don the armband under Allardyce for the first time on Sunday when England launch their attempt to reach the next World Cup in Russia with a game against Slovakia in Trnava.
"Wayne has been an excellent captain for England and the manner in which he has fulfilled the role made it an easy decision for me to ask him to continue," Allardyce said.
"Wayne's record speaks for itself. He is the most senior member of the squad and he is hugely respected by his peers. All of these factors point towards him being the right choice to lead the team."
Rooney was appointed captain by Allardyce's predecessor, Roy Hodgson, in August 2014 after previous skipper Steven Gerrard retired from international football following the World Cup in Brazil.
A year later he broke Bobby Charlton's 45-year-old England scoring record by netting his 50th international goal in a Euro 2016 qualifying win over Switzerland at Wembley.
Rooney's current tally stands at 53 goals and against Slovakia he will become England's most capped outfield player on 116 caps, surpassing David Beckham.
Former goalkeeper Peter Shilton, who made 125 international appearances, is England's most capped player overall.
On the qualities he looks for in a captain, Allardyce said: "I like an on and off-the-field captain, if there is one.
"But if there isn't one, then an on-the-field captain is the main basis of being the captain. You would hope the captain would make a decision on the pitch if he felt he needed to.
"For me, that's the captain's role. It's a big responsibility at club level, even more so at international level. First and foremost, lead by example."
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