Bobrovsky was never drafted into the NHL. Philadelphia Flyers General Manager Paul Holmgren has said that the Flyers considered drafting him during the 2006 NHL Entry Draft, but ultimately decided not to given the difficulty of signing Russian players at the time. After playing his first four professional seasons with Metallurg Novokuznetsk in Russia, Bobrovsky signed a three-year, entry-level contract with the Flyers on 6 May 2010. He was initially slated to begin his North American career with the American Hockey League (AHL)'s Adirondack Phantoms, but after impressing coaches during training camp, Bobrovsky replaced the injured starter Michael Leighton on the roster. Even after Leighton recovered, however, Bobrovsky continued to share duties with Brian Boucher.
Flyers Head Coach Peter Laviolette named Bobrovsky the starting goaltender for the opening game of the 2010–11 season against the Pittsburgh Penguins on 7 October 2010. In his NHL debut, Bobrovsky picked up a 3–2 win and was named the game's first star in the first regular season game at Pittsburgh's Consol Energy Center. At 22 years and 17 days, he became the youngest goaltender to win a season-opening game for the Flyers, eclipsing Ron Hextall, who started at 22 years and 159 days on 9 October, 1986. He was also the first Flyers rookie goaltender to win his NHL debut since Antero Niittymäki did so in 2004. Bobrovsky enjoyed a successful rookie season, posting a 28–13–8 record along with a .915 save percentage. After a disappointing showing during the playoffs, however, in which he was often benched in favor of backup Brian Boucher, the Flyers signed Ilya Bryzgalov to a nine-year contract to be their new starting goaltender.
After a season backing-up Bryzgalov, on 22 June 2012, Bobrovsky was traded to the Columbus Blue Jackets in exchange for Columbus' second round draft choice in the 2012 NHL Entry Draft (Anthony Stolarz), a fourth round pick in the 2012 NHL Entry Draft (Taylor Leier), and the Phoenix Coyotes' fourth round pick in the 2013 NHL Entry Draft (Justin Auger).
As a result of the 2012–13 NHL lockout, Bobrovsky signed a contract with SKA Saint Petersburg of the Kontinental Hockey League (KHL), but returned to the Blue Jackets when the lockout finished. After splitting time early in the season, Bobrovsky supplanted Steve Mason as Columbus's starting goaltender; it was Bobrovsky's first time as a starter since his rookie season in Philadelphia. Bobrovsky was instrumental in the Blue Jackets' push for a playoff spot, which ultimately came up just short, as the team finished in ninth place in the Western Conference. Despite this, however, Bobrovsky had 21 wins, and career bests of a 2.00 goals against average and a .932 save percentage. With this performance, he was awarded the Vezina Trophy, awarded to the league's top goaltender.
On 1 July 2013 he signed a two-year, $11.25 million contract with the Columbus Blue Jackets. On July 3, 2013, he was named to the NHL First All-Star Team.
On 3 December 2013, Bobrovsky was injured in a game against the Tampa Bay Lightning and would be out for four-to-five weeks. He returned to action on 6 January 2014, against the New York Rangers, earning a shootout victory by the score of 4–3. Bobrovsky also made a robbing glove save in the shootout on former Blue Jackets forward star Rick Nash, which was considered to be the game-saving stop.
Bobrovsky backstopped the Blue Jackets in the 2014 Stanley Cup playoffs, where they played the Pittsburgh Penguins in the Eastern Conference quarter-finals. The Blue Jackets won their first playoff game in franchise history on April 19, 2014, in Game 2 by a score of 4–3, and their second-ever playoff victory in Game 4. Bobrovsky played lights out in Game 5, saving 48 of 50 shots, posting a 0.960 save percentage. The Blue Jackets, however, ended up losing the game 3–1. The Blue Jackets were eliminated in Game 6 on April 28, 2014, by a score of 4–3.
During the 2016–17 season Bobrovsky posted a record of 41–17–5, good enough for second in the league while also setting a Blue Jackets franchise record for wins in a season by a goaltender, as well as a 2.06 GAA, .931 save percentage (both of which led the league among qualifying goaltenders), and 7 shutouts (third in the league). He, along with Braden Holtby, were the only goaltenders in the league to be in the top 5 in each category. He also ranked 1st in save percentage within 15 feet (0.896), high stick goals-against percentage (9.4%) and second in 5-hole goals-against percentage (7.9%). For his efforts, he was rewarded with his second Vezina Trophy.
Bobrovsky's regular season success did not carry over to the postseason, however, as he continued to struggle. Throughout a 4–1 series loss to the Pittsburgh Penguins, Bobrovsky won only one game, while posting a 3.88 GAA and an .882 save percentage, with 0 shutouts.
Bobrovsky participated in the 2007 Super Series, an eight-game series between Canadian and Russian juniors commemorating the 35th anniversary of the 1972 Summit Series, and garnered the most ice-time of any goaltender. At the end of that year, he played in the 2008 World Junior Ice Hockey Championships, backstopping Russia to a bronze medal.
Bobrovsky was one of the Russian goalies for the 2014 Winter Olympics in Sochi. He and Semyon Varlamov split time tending the net. After a disappointing loss in a shootout to Team USA, the Russian team would end up being eliminated from the tournament by Finland with a 3–1 loss. Bobrovsky saw some action after Varlamov was pulled for letting in a couple of soft goals in the final game.
Bobrovsky participated in the 2014 International Ice Hockey Federation World Championship in Belarus as Russia's starting netminder. In his first game at the IIHF, he saved all 27 shots, shutting out the Swiss by the score of 5–0. Russia won a berth in the IIHF Finals against Finland, who eliminated Russia from the Sochi Olympics. Bobrovsky faced-off against fellow NHL goaltender Pekka Rinne of the Nashville Predators in the gold medal game. Russia went on to win the gold medal game 5–2, with Bobrovsky saving 24 out of 26 shots, backstopping the Russians to their second gold medal in the past three years in the tournament.
Bobrovsky was named the starting goaltender of Team Russia for the 2015 IIHF World Championship. On 9 May, he saved 13 shots including a penalty shot against Belarus in the group stage to help his team to a 7–0 win. On 16 May in the semifinal, Bobrovsky made several outstanding saves against the U.S., and led his team to a 4–0 win. He eventually finished with 35 saves and was named Player of the Game for Team Russia. On 17 May, he allowed 6 goals on 37 shots in the final, and although he made some great saves early on, it was not enough to keep his team from losing 6–1 to Canada.
Bobrovsky married Olga Dorokova on 16 August 2011. Bobrovsky's nickname is Bob."Badge of Merit for the Town of Novokuznetsk" (21 June 2013) – presented by acting mayor of Novokuznetsk Sergey Kuznetsov
Order of Honor (27 May 2014) – presented by the President of Russia Vladimir Putin for the victory at the 2014 IIHF World Championship as member of Russia men's national ice hockey team
"Order of Honor of Kuzbass" (3 июня 2014) – presented by governor of Kemerovo Oblast Aman Tuleyev for the skills and professionalism in sports activities
NHL All Star selection: 2015, 2017
NHL Rookie of the Month: November 2010
NHL's First Star of the Week: Weeks ending 10 March 2013, 14 April 2013, 16 March 2014
NHL's Second Star of the Week: Week ending 6 March 2017
NHL's Third Star of the Week: Week ending 17 March 2013
NHL's First Star of the Month: December 2016
NHL's Second Star of the Month: March 2017
NHL's Third Star of the Month: March 2013
Most wins in a season: 41 (2016–17)
Best goals against average in a season: 2.00 (2012–13)
Best save percentage: .932 (2012–13)
† Could not play due to injury.