Essiedu scooped best performance in a play at the UK Theatre Awards, which reward shows produced outside London.
The 26-year-old played Shakespeare's prince in Stratford-upon-Avon.
Two musicals from Sheffield's Crucible Theatre - Flowers For Mrs Harris and Show Boat - led the other winners, while Vanessa Redgrave and Sir Ian McKellen received honorary awards.
Essiedu said his award was "a huge honour" and that the fact he was the RSC's first black Hamlet was "significant and insignificant at the same time".
He said: "It's significant in that the RSC is a national organisation that should represent everyone in our country regardless of your creed, colour, religion, sexuality, whatever.
"As such you would expect something like this to have happened before. But I think it was really important that it was happening now and we were really proud of the work that we did."
Sir Ian McKellen was given an award for his outstanding contribution to British theatre and told the ceremony about the importance of his visits to the three theatres in his home town of Bolton in his youth.
"If it hadn't been for those local theatres, where I did my first theatregoing, I wouldn't be an actor, of course not," the 77-year-old said.
The number of theatres around the country has declined since then, he said - but he added that the standards in those that now exist, like Bolton's Octagon, were often higher than in the past.
Vanessa Redgrave was presented with the Gielgud Award for excellence in the dramatic arts and told the audience she believed theatre "helps keep society sane".
"And boy do we need to be kept sane at the moment. And if I've helped along the way," she added with a laugh, "I'm glad."
Among the other winners, Flowers For Mrs Harris and Show Boat, both directed by Sheffield Theatres' former artistic director Daniel Evans, shared the award for best musical production.
Clare Burt won best performance in a musical for Flowers for Mrs Harris, a new musical that was adapted from a 1958 novel by Paul Gallico.
Rebecca Trehearn picked up best supporting performance for playing Julie La Verne in Sheffield's production of classic musical Show Boat, which later transferred to the West End.
The prize for best new play went to Cuttin' It by Charlene James, a drama about female genital mutilation among teenage girls in the UK.
An operatic adaptation of Sarah Kane's innovative and intense play 4:48 Psychosis by the Royal Opera and Guildhall School Of Music and Drama earned the prize for achievement in opera.
Choreographer Gary Clarke, who picked up the achievement in dance accolade for Coal, his show about the 1980s mining industry, dedicated it to his home town of Grimethorpe, South Yorkshire, and other mining communities.
And a theatre in a restored 18th Century flour mill on the outskirts of Reading, The Mill at Sonning, was voted the UK's most welcoming theatre.
The UK Theatre Awards do not cover National Theatre productions or West End shows unless they have transferred from a theatre outside the capital.
The awards were handed out at a ceremony at London's Guildhall on Sunday.
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Paapa Essiedu, the first black actor to play Hamlet for the Royal Shakespeare Company, has won a top theatre award.
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