Dawson was born on 16 July 1863 at Rockhampton, Queensland, the son of Andrewson Dawson and his wife Jane (née Smith). When he was six his mother died and he was placed in Diamantina orphanage in Brisbane until he was nine, when an uncle took him to Gympie. He began work as a miner at Charters Towers, and later was elected first president of the Miners' Union. He took up journalism and for a time was editor of the local newspaper, The Charters Towers Eagle.
Dawson entered politics at the 1893 election, as one of the two Labor candidates for Charters Towers in the Legislative Assembly of Queensland. He won the seat, and retained it at the 1896 and 1899 elections. When the government of James Dickson resigned on 1 December 1899, Dawson formed a ministry. Although it was defeated as soon as the Legislative Assembly next met, it nevertheless became the first socialist or Labour Party government in the world.
At the first Federal election for the Senate in 1901, Dawson was returned at the head of the Queensland Labor ticket. While in federal parliament, he was regarded as a good speaker, but struggled with persistent ill health associated with chronic lung trouble from his time as a miner, which worsened after he relocated his family from Queensland to the colder climate of Melbourne. He also struggled with alcoholism, and was absent from parliament for periods, which frustrated his colleagues.
In April 1904, when Chris Watson formed the first Federal Labor government, Dawson was given the portfolio of Minister for Defence in light of his prominent status as a former Premier. As Minister for Defence, he clashed with Edward Hutton, the aristocratic English General Officer Commanding the Australian Military Forces, who had resisted being answerable to the executive, and had been viewed as disrespectful by successive defence ministers. Dawson proposed a military restructure which eliminated Hutton's position, which was adopted by his successor after the ousting of the Watson government, resulting in Hutton's resignation and return to England. Dawson reportedly stated that the "most satisfying facet" of his stint as minister had been that he had "pulled down from his pedestal the biggest bounder that had ever commanded the forces in Australia."
By the 1906 election, Dawson had a poor relationship with the Queensland state executive of the Labor Party, and was initially demoted to the unwinnable fourth position on the Labor Senate ticket. As a result of concerns about the electoral fallout of his dumping, he was reinstated to the winnable third position on the ticket, but resigned as a candidate two months later, citing ill health. He subsequently changed his mind, but the executive refused to reinstate him, so he ran as an independent. That move split the Labor vote, and amidst a generally bad election for Labor in Queensland, the entire ticket lost.
Dawson was unable to find work in Melbourne, and returned to Queensland in 1909, while his wife and four children remained in Melbourne. He was admitted to the Brisbane General Hospital on 6 July 1910 and was expected to recover, but died of the effects of alcoholism on 20 July 1910. His widow and children reportedly did not attend his funeral. He was buried in Toowong Cemetery on 21 July 1910.
The Federal electoral division of Dawson is named after him.