He was originally selected by Salavat Yulaev Ufa in the first round, 7th overall, of the 2011 KHL Junior Draft.
Vasilevskiy was selected 19th overall in the 2012 NHL Entry Draft by the Tampa Bay Lightning. On 6 May 2014, he was signed by the Lightning to a three-year, entry-level contract.
On 27 September 2014, Vasilevskiy was assigned to the Syracuse Crunch of the American Hockey League (AHL), the top minor league affiliate of the Tampa Bay Lightning. On 15 December 2014, Vasilevskiy was named the CCM/AHL Player of the Week. In the previous week, he allowed just one goal on 56 shots faced in two games, which was good for a 0.50 goals against average (GAA) and a .982 save percentage. Vasilevskiy had a 29-save shutout against the Springfield Falcons, which snapped their franchise-record 11-game winning streak. In the next game, he made 29 saves in a 4–1 victory over the Hershey Bears. Vasilevskiy has compiled a record of 8–3–3 in 14 appearances with Syracuse this season, with a 2.34 GAA, .918 save percentage and two shutouts. On 2 January 2015, he was named CCM/AHL Goaltender of the Month for December, posting a 4–0–1 record with a 1.17 GAA and a .962 save percentage. He is the first Syracuse goaltender to win the award since Karl Goehring was honored in March 2008.
On 16 December 2014, Vasilevskiy was recalled to the Tampa Bay Lightning. He then played and started later that night in his NHL debut, a 3–1 victory over the Philadelphia Flyers. He also accomplished another feat that night by becoming the first NHL goaltender to ever wear the number 88. He recorded his first NHL shutout on 3 March 2015, over the Buffalo Sabres in a 28-save performance. Vasilevskiy's first playoff win came on 6 June 2015, when he replaced starting goaltender Ben Bishop twice during the third period of Game 2 of the Finals; he thus became the first goalie in 24 years to win a playoff Final in relief and the first to earn his first career playoff win in relief in the Final since Lester Patrick in 1928.
On 3 September 2015, Vasilevskiy had successful surgery to remove a blood clot from near his left collarbone and to treat a type of Vascular Thoracic Outlet Syndrome. Vasilevskiy was expected to return to the ice in 2–3 months. On 21 October 2015, Vasilevskiy was cleared to come off of his blood thinner medication, allowing him to return to practice. On 1 November 2015, the Lightning recalled Vasilevskiy from his conditioning stint with the Crunch, where he made 56 of 58 saves in two wins. Vaslievskiy made his return to the NHL that same night, in a 4–3 Lightning victory over the Carolina Hurricanes.
On July 1, 2016, the Lightning announced the re-signing of Vasilevskiy to a three-year contract extension. Vasilevskiy appeared in 24 games with the team last season, posting a record of 11–10–0 to go along with a 2.76 goals-against average and .910 save percentage. He had one shutout in his second season with the team. He also played in eight Stanley Cup playoff games in 2016, recording a record of 3–4 to go along with a 2.76 goals-against average and .925 save percentage. On December 28, 2016, Vasilevskiy recorded his first career NHL point as an assist on an overtime goal scored by Tyler Johnson, which came in a 4–3 victory over the visiting Montreal Canadiens.
In winning the 2014 IIHF World Championship with the Russian senior team, Vasilevskiy was awarded the Order of Honour on 27 May 2014.
On 2 March 2016, the Russian Ice Hockey Federation named Vasilevskiy to its roster for the 2016 World Cup of Hockey. Vasilevskiy is being joined by Lightning teammates Vladislav Namestnikov, and Nikita Kucherov. The tournament took place from 17 Sep to 1 October 2016, in Toronto.
On April 9, 2017, the Russian Ice Hockey Federation named Vasilevskiy to its roster for the 2017 World Ice Hockey Championships. Vasilevskiy was joined by teammate Nikita Kucherov. On May 21, 2017, Vasilevskiy helped Russia capture a bronze medal when they defeated Finland in the bronze medal game. Additionally, Vasilevskiy was voted the top goaltender of the tournament.
His father, Andrei Vasilevski (born 1966), was also a goaltender who competed in the Russian Superleague as a member of Salavat Yulaev Ufa. Vasilevski is married and has a son.