Unsworth, the son of Joseph and Olive Unsworth, was born in Dubbo, New South Wales, and educated in Sydney, at Kogarah High School. On leaving school at age 15, he was apprenticed as an electrical fitter. In 1955, aged 21 years, he married Pauline Hennessy and they subsequently had one daughter and three sons, one of whom has died; he and his wife have nine grandchildren, and a great grandchild. Unsworth was initially an Apprentice Electrical Fitter, then Electrical Fitter, Electrical Testing Officer and subsequently Sales Representative for Sydney County Council from 1950 until 1960. Unsworth had a brief period of military national service in 1953-1954 in the Royal Australian Navy.
Unsworth was elected a member of the New South Wales Legislative Council in 1978. While serving in the Legislative Council, Unsworth represented Australia as a delegate at the International Labour Organization Worker Participation in Management Conferences held in Geneva and The Hague. In the Wran Labor Government, Unsworth was Minister for Transport and Minister for Health. After the surprise retirement of Premier Neville Wran in May 1986, he became leader of the NSW Labor Party and thus Premier, and was also Minister for Ethnic Affairs and Minister for State Development. As Premiers are required by convention to be members of the Legislative Assembly, Brian Bannon, the member for the normally safe Labor seat of Rockdale, resigned to accept a government job as Chairman of the Homebush States Sport Centre Trust, and Unsworth contested the resulting by-election held on 2 August 1986. He only narrowly won the seat, with a 17.1% dive in the primary vote and hostile independent preferences giving him a margin of just 54 votes. A by-election for Wran's equally safe seat of Bass Hill was even worse, with a 22.2 per cent drop on the primary vote delivering a 103-vote victory to the Liberal candidate. Unsworth was Premier for two years until the Labor Party's landslide defeat by the Liberals' Nick Greiner in the 1988 elections. Unsworth did not contest the 1991 election.
Life after politics
Immediately following politics, Unsworth had little political involvement with his only public commitment being as Deputy Chairman and then Chairman of the Australian Executive Committee for His Royal Highness Duke of Edinburgh Study Conference. Unsworth was General Manager of 2KY Broadcasters Pty Ltd Racing Radio 1992–2000. In 2001 he was a recipient of the Centenary Medal. Following the election of the Carr Labor Government in 1995, Unsworth was placed on a number of key government and community bodies, including:
During 2003 - 2008, Unsworth was a Director of Father Chris Riley's charity, Youth off the Streets, that provides crisis care, refuges, schools, drug programs and alcohol programs to young people in Australia and throughout Asia. Unsworth has also been a Director of Entherm Pty Ltd, Member of Overseas Trade Authority of New South Wales and a Director of Manly-Warringah Sea Eagles. In 2003 and 2004, Unsworth was commissioned by the NSW Government to conduct a formal review of public bus services. In 2008 he headed a committee that considered the privatisation of the New South Wales' electricity industry. The committee and Unsworth supported the sale.