The Big Finish Productions audio drama - starring Paul McGann and Toby Jones among others - won the Best Online or Non-Broadcast Audio Drama prize at the annual ceremony.
Lenny Henry hosted the event in the radio theatre at BBC's Broadcasting House in London on Sunday (January 26).
Elsewhere, 94-year-old June Spencer won a Lifetime Achievement honour for her longrunning role of Peggy Woolley in BBC Radio 4's The Archers.
Spencer has played the role since the soap began in 1951, and was recently at the centre of a storyline about the death of her on-air husband (Arnold Peters), who had been suffering from dementia.
Peters himself was diagnosed with Alzheimer's disease, and passed away last May. Spencer also lost her real-life husband to the disease.
The Archers editor Sean O'Connor said: "June is an extraordinary woman who has committed her life to the BBC and The Archers. At 94, absolutely nothing fazes her.
"She performed with heart-breaking dignity in the recent episodes, using her instincts as an actress and channelling her own difficult experiences of caring for a loved one with Alzheimer's.
"It's a testament to her that these episodes have managed to move listeners so profoundly."
Other winners included Best Actor for Lee Ross in Radio 3's King David.
A Radio 4 production of Evelyn Waugh's Sword of Honour trilogy was named Best Audio Drama Adaptation.
John Finnemore's Cabin Pressure - starring Benedict Cumberbatch - missed out in Scripted Comedy and Scripted Comedy Drama categories.
BBC director general Tony Hall said: "The BBC is unique in realising the power of radio drama. It is something we have believed in for 90 years - and continue to champion. Britain is famous for the quality of its writers, actors and directors. My congratulations to everyone involved."