Prior to becoming a professional footballer, Dawson worked as a glass collector at his local pub, while playing for a local team called Rochdale St Clements. He started his football career at Northern Premier League side Radcliffe Borough in the middle of the 2007–08 season, and became a first team regular after just a few games. In the two seasons he was at the club he made 95 appearances and scored 15 goals. Dawson signed for his local club Rochdale in February 2009 on a two-year contract. Although a Rochdale player, Dawson stayed at Radcliffe on loan until the end of the season. In the season he left, he was voted the Player of the season by Radcliffe. In 2017 Craig was included in the Northern Premier League’s Greatest 100 Players for the 50th Anniversary of the league, ranking in 14th overall

On 31 August 2010, Dawson signed for Premier League club West Bromwich Albion for an undisclosed fee on a three-year contract. He made his competitive Baggies debut in a 4–1 League Cup win at Bournemouth in August 2011 and the following day was called up into the England Under-21s squad by Stuart Pearce, who watched him from the stands at the Seward Stadium. On Tuesday 26 July 2011, Craig Dawson was rewarded with a new-and improved Albion contract on 21 February 2012. The centre-half penned a three-and-a-half-year deal to June 2015, plus a further year’s option in the club’s favour. Dawson made his first Premier League start on Saturday 17 September 2011, away from home against Swansea City as a fill-in central defender for the suspended Gabriel Tamaş in a disappointing 3–0 defeat. He re-appeared in the starting eleven for the game against Manchester City in which West Brom went on to lose 4–0. Again, Dawson appeared in the next league game against Queens Park Rangers, West Brom went on to win the game 1–0. On 28 September 2014, Dawson scored his first Premier League goal for West Brom in a 4–0 home win against Burnley He scored his next goal of the season in a 1–2 loss to West Ham United on 2 December 2014.

Dawson scored two goals on his England under-21 debut against Azerbaijan on 1 September 2011. He played 15 matches for England U21s, scoring six goals.

On 2 July 2012, Dawson was selected by Stuart Pearce for the 2012 Great Britain Olympic football team. He featured in the 1–0 victory over Uruguay as a last minute substitution for goalscorer Daniel Sturridge. He also featured in the Olympic quarter-final against South Korea, coming on as a substitute for an injured Micah Richards, and scoring in the penalty shoot-out.

On 23 January 2013, despite interest from Blackburn Rovers, Leicester City, Leeds United and Nottingham Forest, championship club Bolton Wanderers won the race to sign Dawson on loan for the rest of the season. On 23 February, Dawson scored twice in a 4–1 win against Hull City. Just three days later on 26 February, he scored the only goal in a 1–0 home win against Peterborough United. He scored for the third successive match on 2 March, scoring the winning goal in a 3–2 win against Barnsley.

After the arrival of new Albion manager Tony Pulis at the start of 2015, Dawson became a regular feature in the starting lineup. While not chipping in with any more goals that season, Dawson played his part in the team achieving survival. During the 2015/16 season Dawson was again a regular starter. His first goal of the season came in a 2–3 loss to Everton on 28 September 2015. His next goal came in a 2–2 draw away at Liverpool on 13 December 2015. On 18 February 2016 he signed a contract extension to keep him at the club until the summer 2018. Two weeks after signing his contract extension he scored a goal in a 3–2 home win against Crystal Palace. He was also involved in the winning goal for Saido Berahino. On 25 April 2016 he became the fourth Premier League player of the season to score an own goal and a goal in the same match in a 1–1 draw away to Tottenham.

On 18 March 2017, he scored two headers from corners, as West Brom defeated Arsenal 3–1 in the Premier League, he was named man of the match. Dawson made 28 league appearances for West Brom in the 2017–18 season as they were relegated from the Premier League. He subsequently handed in a transfer request and refused to travel to Portugal for the club’s training camp.

On 1 July 2019, Dawson signed for Premier League club Watford for an undisclosed fee on a four-year contract.

Dawson scored his first goal for the club on 20 June 2020, An overhead kick against Leicester City. Dawson made 29 appearances for Watford in the 2019-20 season as they were relegated to the Championship. At the start of the 2020–21 season, and the appointment of new manager, Vladimir Ivić, his selection for the team became less likely following the signings of Francisco Sierralta and William Troost-Ekong.

On 12 October 2020, Dawson signed for West Ham on loan for the remainder of the 2020–21 season. The club retained an option to make the transfer permanent at the end of the loan.