He is perhaps best known in the antiques trade for his portrayal of Tinker Dill in the series Lovejoy, which ran from 1986 to 1994.
Sutton (1933-2018) died at the Royal Trinity Hospice in Clapham, south London at the weekend.
A message from his family said: "Today we're devastated by the loss of our beautiful Dudley, who leaves a gaping hole in all our lives.
"We're grateful for the love expressed by friends and fans everywhere and for the extraordinary care he received at the [hospice], where he went out fighting for our NHS."
Actors and friends led the tributes on social media including Robert Lindsay and fellow Lovejoy star Chris Jury, who played Eric Catchpole in the BBC series.
Oh my darling Dud what a great career what a great person https://t.co/92wSlDEjAJ— Robert Lindsay (@RobertLindsay) September 15, 2018
Many in the antiques trade also paid their respects via Twitter.
Still my absolute favourite ever TV characters suit.— Brown Bowler (@brownbowlerhat) September 15, 2018
Farewell Tinker... pic.twitter.com/IauFKgO1Eo
Sad news....what an acting career. He will always be Tinker to me, a fantastic character, so perfectly played. For Tinker the gavel has fallen, the auction is over. Bless you. https://t.co/8Qw6UqsT07— Mark Ashley (@markmashley1) September 16, 2018