Smith died on Tuesday at a clinic near the South African city of Cape Town, where he spent his final years, according to long-time friend Sam Whaley.
He had been ailing for some time and recently suffered a stroke.
To many white Rhodesians, he was a kind of idol, but for most blacks, his rule symbolised the worst of racial oppression.
A former Royal Air Force pilot, he fought for Britain in the second world war, then rebelled against it 20 years later.
Smith imprisoned Mugabe, Zimbabwe's current president, in 1964 for 10 years, calling him a "terrorist" intent on turning the country into a one-party dictatorship.
"We have never had such chaos and corruption in our country," Smith said during a brief return to politics in 2000.
"What Zimbabweans are looking for is a bit of ordinary honesty and straightforwardness."