He started playing cricket at the age of 3 and joined Goatacre when he was 7. He then moved on to play his youth cricket at Chippenham Cricket Club in Wiltshire, where he was spotted by Hampshire.

After appearing regularly for Hampshire Second XI and Wiltshire in 2006 he was selected for England under-19s tour of Malaysia in 2006/07. During the tour he took impressive figures of 6/9 against Malaysia. During England under-19s 'Test' series with Pakistan in 2007 he was England's leading wicket-taker.

Dawson appeared in three List A matches for Hampshire towards the end of the 2007 season, he failed to take a wicket but he scored a run-a-ball 32 on his debut. On 19 September 2007 he made his first-class debut against Yorkshire, although he failed to bat or bowl.

He scored his maiden century at Trent Bridge against Nottinghamshire in 2008. In January 2009 he was called into the England Lions squad to tour New Zealand. Dawson was a member of Hampshire's 2009 Friends Provident Trophy winning team. In 2010 he played just eight Championship matches, averaging 29 with the bat. In 2011, he had a spell opening the batting for Hampshire. He averaged 36 in the Championship. He averaged 35 in List A games as he began to build a reputation for himself as a solid batsman.

In 2012 his bowling began to become more prominent, as he picked up 26 wickets for Hampshire in the County Championship, averaging 32. However, his batting became less prominent as he averaged less than 30 for the season. He also became an important part of Hampshire’s bowling line-up in the T20, taking nine wickets.

In 2013 he scored over 1,000 first-class runs for the first time in his career, although he was less effective with the ball, taking just eleven Championship wickets. However, his bowling in One Day cricket became more effective as he took 12 List A wickets, the most in his career. He continued his impressive performances with the ball in T20 cricket, taking a further 13 wickets. In 2014, after falling out of favour at Hampshire, Dawson moved to Essex on loan. Dawson performed well during the loan, and when he returned to Hampshire he reclaimed his spot in the side. In 2015 he took 29 wickets in the Championship, and 12 in List A cricket as his bowling continued to improve.

In 2016 Dawson earned his first call up to a senior international squad when he was selected for England's squad for the 2016 ICC World Twenty20, although he didn’t play a game for the side.

On 5 July 2016 he made his Twenty20 International (T20I) debut for England against Sri Lanka. He took 3 wickets as England won by eight wickets.

On 4 September 2016 he made his One Day International (ODI) debut for England against Pakistan. He scored ten runs and taking figures of 2-70 as England lost by four wickets.

In November 2016 he was named in England's Test squad for the final two matches of the series against India. He made his Test debut on 16 December 2016 against India in the fifth Test of the series. He scored an unbeaten 66 in his maiden innings, as England made 477.^[5] Murali Vijay became his first test wicket, after he trapped him lbw, as he finished with figures of 2-129, before being dismissed for a duck, as England lost by an innings and 75 runs.

Dawson played in the second T