A government statement said Dlamini, 52, died on Sunday afternoon in hospital in South Africa.
No cause of death was mentioned, but Dlamini had recently been receiving treatment for Covid-19 in South Africa.
Dlamini had been prime minister of Eswatini, formerly known as Swaziland, since October 2018.
The tiny landlocked country in southern Africa is one of the last absolute monarchies in the world.
With a population of about one million, the country has recorded 6,768 coronavirus infections and 127 deaths linked to the pandemic, according to the health ministry.
Dlamini announced he had tested positive for coronavirus on 16 November. At the time, he said he was asymptomatic and was feeling well.
A few weeks later, on 1 December, the Eswatini government said Dlamini had been moved to a hospital in South Africa, with the aim of fast-tracking his recovery.
But on Sunday the government said Dlamini died “while under medical care in a hospital in South Africa”, without giving more details.
“Government in collaboration with the family will keep the nation informed of subsequent arrangements,” Deputy Prime Minister Themba Masuku said in the statement.
Dlamini was among a number world leaders to have contracted Covid-19 during the pandemic, including US President Donald Trump, UK Prime Minister Boris Johnson and Brazilian President Jair Bolsonaro.