The You, Me and the Big C podcaster, who also set up the Bowelbabe cancer research fund, died last year aged 40.
She made a touching last appearance alongside Livesey on BBC Radio 5 Live last May, shortly before her death.
It was named the Radio Times Moment of the Year at the Aria Awards on Tuesday.
In the interview, Livesey was heard refusing to say goodbye, while thanking Dame Deborah and offering "a big warm hug".
She responded by thanking him back for his support, as well as listeners for "coming on this crazy journey with us."
Other winners included comedian Romesh Ranganathan - for his BBC podcast For the Love of Hip Hop - as well as BBC radio DJs Craig Charles and Tony Blackburn.
Blackburn won the pioneer award shortly before being discharged from London's Wellington hospital, where he had been for three weeks with a chest infection.
"After my treatment and a rest I look forward to getting back at Radio 2 and BBC local radio to doing what I love doing," he wrote on Twitter.
Elsewhere at the awards, run by the Radio Academy, 1Xtra's Nadia Jae picked up the prize for best music breakfast show. 5 Live Breakfast won best speech breakfast show and TalkSport's Adrian Durham was named best speech presenter.
Radio 4 won UK station of the year, and In Dark Corners - its investigation into historical sexual abuse at Scottish schools - was named best factual series.