Before going to North America, Mrázek played in the Czech Extraliga, where he appeared for HC Vítkovice Steel during the 2007–08 post-season.

In his first Ontario Hockey League (OHL) season with the Ottawa 67's in 2009–10, Mrázek posted a 12–9–1 record in 30 games with a 3.00 goals against average (GAA) and .905 save percentage.

At the end of the season, at the 2010 NHL Entry Draft from 25 and 26 June, Mrázek was selected by the Detroit Red Wings in the fifth round, 141st overall.

On 4 April 2011, upon the conclusion of Ottawa's season, Mrázek was signed by the Grand Rapids Griffins, the Detroit Red Wings' American Hockey League (AHL) affiliate, on an amateur try-out contract.

Mrázek posted a 2–1–0 record, a 2.02 GAA and a .944 save percentage in three games played in the ECHL before being called up to Grand Rapids in the AHL.

[8] During the 2014–15 season, he appeared in 29 games for the Red Wings, posting a 16–9–2 record, 2.38 GAA, .918 save percentage and three shutouts.

Since being assigned to the Griffins on 17 February, Mrázek has not suffered a regulation loss, posting a 6–0–1 record, with a 1.13 goals against average, a 0.962 save percentage and three shutouts.

Only Washington's Braden Holtby (.932) and Florida's Roberto Luongo (.931) have better save percentages among goalies who have logged as many games played.

On April 4, 2019, Mrázek earned his 23rd win of the season as the Hurricanes clinched a playoff berth for the first time in 10 years.

Mrázek and the Hurricanes then upset the defending Stanley Cup champion Washington Capitals in an opening round seven-game series en route to the Eastern Conference Finals, also for the first time in 10 years.

He suffered an injury during the second round against the New York Islanders and was replaced by goaltender Curtis McElhinney for several games.

[23] On July 28, 2021, Mrázek signed as a free agent to a three-year, $11.4 million deal with the Toronto Maple Leafs.

[24] Maple Leafs general manager Kyle Dubas envisioned him operating as part of a goaltending tandem with Jack Campbell.

Both Red Wings goaltenders also played poorly, leading to a 10–7 Leafs victory, one of the highest-scoring games in recent NHL history.

[31] Mrázek represented the Czech Republic at the 2012 World Junior Ice Hockey Championships in Alberta, Canada.

He backstopped the Czech Republic to the quarterfinals, where his team was finally eliminated from medal contention in overtime by the Russians.

[6][32] Mrázek represented the Czech Republic at the 2012 IIHF World Championship as the third goaltender, where he won a bronze medal.