Hazelwood began kart racing at age seven at the Adelaide Dirt Kart Club, progressing through junior divisions to win multiple state and national karting titles on both dirt and bitumen. He transitioned to Formula Ford in 2011 and competed in Australian Formula 3 in 2013 with R-Tek Motorsport, finishing second in the championship and recording his first pole position at Sydney Motorsport Park.
Hazelwood joined the Dunlop Super2 Series — the Supercars feeder championship — with Matt Stone Racing in 2014 in a Ford FG Falcon, finishing eighth overall as the highest-placed rookie and winning the Mike Kable Young Gun Award. He finished fourth in 2015 and third in 2016, winning the Supercars Privateers Cup in the latter year. His dominant 2017 campaign produced 14 front-row starts, 13 podiums, six race wins, and three pole positions, earning him the championship by 95 points over Paul Dumbrell.
Hazelwood graduated to full-time Supercars competition with Matt Stone Racing in 2018, inheriting an ex-DJR Team Penske Ford Falcon and initially struggling before a mid-season switch to the VF Holden Commodore the team had used in Super2. Results improved significantly in 2019 after MSR acquired an ex-Jamie Whincup Holden ZB Commodore and formed a technical alliance with Triple Eight Race Engineering, with Hazelwood recording a career-best fifth at Pukekohe.
He joined Brad Jones Racing for 2020, achieving a breakthrough season: he recorded his maiden podium (third at Sydney Motorsport Park) and maiden pole position (Townsville), and was genuinely competitive for race wins on several occasions. BJR retained him for 2021, where he recorded a personal-best Bathurst 1000 finish of eighth with Dean Fiore and amassed 12 top-ten results across the season.
After a return stint with Matt Stone Racing in 2022 and a season with the Blanchard Racing Team in 2023, Hazelwood stepped back to a co-driver role with Erebus Motorsport for 2024. He filled in early in the season for championship Brodie Kostecki before pairing with him for the endurance rounds.
After a disappointing Sandown 500 retirement, Hazelwood and Kostecki arrived at Bathurst with motivation to respond. Kostecki qualified on pole and the pair dominated the race from start to finish, winning in a record time of 5 hours and 58 minutes — the fastest Bathurst 1000 in history, the first completed in under six hours, aided by just one safety car period. It was Hazelwood's maiden Supercars race win and made him the 66th different winner of the Bathurst 1000.
Hazelwood debuted in the Trico Trans Am Series in 2024 with TFH Racing, winning seven races, two pole positions, and the series title in his debut season — defeating the dominant Garry Rogers Motorsport operation. He returned to defend the title in 2025.
For 2025, Hazelwood joined Dick Johnson Racing as co-driver for Brodie Kostecki in the Shell V-Power Ford Mustang, recording a win at The Bend 500 and a pole position at the Repco Bathurst 1000.