Vaughan, who had been in charge at the organising committee since early last year, "stepped down" by "mutual consent", according to a statement from organisers England Rugby 2015.
Jevans, who was formerly general secretary of the International Tennis Federation, will take up her new role in November after she has wrapped up her work at the ongoing London Paralympics.
"I have spent the last decade planning and delivering the world's two largest sporting events," she said. "This has been an extraordinary journey and I am looking forward to putting my experience and expertise in to delivering an exceptional Rugby World Cup in 2015."
With just three years to go until the tournament begins, the first major tasks for Jevans will be to select the venues, formulate the match schedule and develop the ticketing programme.
The appointment of Jevans was also welcomed by the International Rugby Board, which owns the rights to rugby's showpiece event through Rugby World Cup Limited.
"The successful delivery of one of the world's largest sporting events is founded on a clear vision, partnership, teamwork and strong leadership," IRB and RWCL chief Bernard Lapasset said in a statement.
"As a driving force behind the exceptional London 2012 Olympic and Paralympic Games, Debbie certainly possesses all the necessary attributes to drive the delivery of what promises to be a special tournament."