He has previously played in the NHL with the Nashville Predators, New York Rangers, Ottawa Senators, Detroit Red Wings, and Pittsburgh Penguins.

[4] On 26 October, Hellberg made his NHL debut with the Predators, playing 12 minutes after replacing starter Carter Hutton in a game against the St. Louis Blues.

[5][6] After being returned to Milwaukee, Hellberg suffered a high ankle sprain when his skate caught under a goal post which caused him to miss three months.

[10] Hellberg was assigned to the Rangers AHL affiliate, the Hartford Wolf Pack, upon his arrival in New York but was called up on 18 December[11] to serve as a backup to Henrik Lundqvist while Antti Raanta was recovering from a head injury.

[11] Hellberg was called up to the NHL three times during the 2016–17 season; the first call-up came on 15 January 2017 after Raanta suffered a lower-body injury against the Montreal Canadiens.

[17] As an impending free agent, Hellberg left the NHL and signed a one-year contract to be the starting goaltender of Chinese club, Kunlun Red Star of the Kontinental Hockey League (KHL), on 25 May 2017.

Having earlier signed a one-year extension with Kunlun during the season, Hellberg was traded the following off-season by the Red Star to perennial contending club, SKA Saint Petersburg, in exchange for financial compensation on 28 May 2018.

[19] After three stellar seasons with SKA Saint Petersburg, Hellberg left the club at the conclusion of his contract and opted to continue in the KHL by agreeing to a one-year deal with HC Sochi on 15 June 2021.

Unable to help Sochi qualify for the postseason, Hellberg signed a one-year contract in a return to the NHL for the remainder of the Detroit Red Wings' season on 13 April 2022.

[25] Hellberg served as the backup to Martin Jones through several games with the Kraken before he was placed on waivers after the return to health of goaltender Philipp Grubauer on 22 November.

[36] On 8 March 2024, the Penguins traded Hellberg to the Florida Panthers in exchange for goaltender Ludovic Waeber and a conditional 2025 seventh-round pick.

[38] Leaving the Panthers as a free agent without featuring for the club, Hellberg continued his career in North America after agreeing to a one-year, two-way contract with the Dallas Stars on 13 August 2024.