At age four, Horvat played for The West Lorne Comets. At age 12, Horvat played peewee for the Toronto Red Wings, winning the all-Ontario Peewee AAA championships in 2008. With ten points in the round-robin portion of the tournament, he won the leading scorer award. Living more than 200 kilometres away from Toronto, Horvat was driven into the city every weekend by a teammate’s father and practised separately from the team throughout the week.

By 2010, Horvat advanced to the midget level with the Elgin-Middlesex Chiefs. He scored 47 goals and 118 points over 68 games in his first and only season with the team and was named the Minor Hockey Alliance of Ontario’s midget player of the year. At the end of the season, he played an additional five games of Junior B for the St. Thomas Stars, a team coached and owned by his uncle, Ron Horvat (Horvat’s father also helped out with the team). During his brief stint with St. Thomas, he registered a goal and four points.

In the summer following his midget season, Horvat was selected 9th overall by the London Knights in the 2011 Ontario Hockey League (OHL) Priority Selection Draft. Joining the Knights for the 2011–12 season, he scored a goal in his OHL debut on September 23, 2011, an 8–0 win over the Saginaw Spirit. Horvat completed his OHL rookie season with 11 goals and 30 points in 64 games, ranking 18th among the league’s first-year players. In the playoffs, he contributed a goal and three assists over 18 games, as London won the J. Ross Robertson Cup as league champions. With the Knights advancing to the 2012 Memorial Cup, Horvat scored a goal in three games. London finished as tournament runner-up to the Shawinigan Cataractes, losing 2–1 in the final.

Entering the 2013 NHL Entry Draft, Horvat was ranked 15th among North American skaters by the NHL Central Scouting Bureau, while International Scouting Services ranked him 10th overall among all draft-eligible skaters. He was scouted as a two-way forward who could contribute in all areas of the game, both offensively and defensively. When asked to compare himself to an NHL player, Horvat stated that he tries to model his game after Mike Richards, a Selke Trophy nominee in previous years. Horvat was taken ninth overall in the draft by the Vancouver Canucks, who had traded goaltender Cory Schneider to the New Jersey Devils in exchange for their first-round draft pick.

While attending Canada’s national junior summer camp, Horvat was signed to a three-year, entry level contract on August 6, 2013. Following his first training camp and pre-season with the Canucks, Horvat appeared to have made the team’s opening roster. After the Canucks made a trade for a pair of centres from the Carolina Hurricanes, however, he was reassigned to the OHL for another season of junior.

Horvat made his NHL debut on November 4, 2014, against the Colorado Avalanche. Six games later, he scored his first NHL goal on November 20 against Frederik Andersen of the Anaheim Ducks. The following game, he registered his first three assists in a 4–1 win against the Chicago Blackhawks. On November 25, the Canucks announced that they would not be returning Horvat to the OHL despite his remaining year of junior eligibility. Despite Horvat’s status as a highly anticipated prospect, Canucks head coach Willie Desjardins admitted later in the season he had not expected Horvat to compete for a roster spot. Playing on a line with wingers Jannik Hansen and fellow rookie Ronalds Ķēniņš, Horvat earned bottom-six ice time in his first year with Vancouver, averaging 12:15 minutes per game. In 68 games, Horvat scored 13 goals and 12 assists. His regular season play earned him two fifth-place votes for the Calder Memorial Trophy, ranking him 11th among first-year players for the rookie of the year award. During the playoffs, he added a team-high four points in six games during the Canucks’ first-round loss to the Calgary Flames.

In 2015, the Vancouver Canucks had the option of loaning Horvat to Team Canada for the World Juniors, but instead chose to retain him in the NHL.

On January 10, 2017, Horvat was named an All-Star for the 2017 NHL All-Star Game after having tallied 13 goals and 16 assists up to that point. Horvat was the youngest Canuck to be selected for an All-Star Game since Trevor Linden. On March 3, 2017, Horvat was named as one of Vancouver’s second alternate captains, replacing longtime Canuck Alex Burrows. Horvat ended the 2016–17 season with 20 goals, 32 assists and 52 points, setting career highs in all three categories.

On September 8, 2017, following the expiry of his entry-level contract after the previous season, the Canucks re-signed Horvat to a six-year, $33 million contract extension worth $5.5 million annually. On December 5, 2017, Horvat suffered a broken foot during a game against the Carolina Hurricanes which resulted in him missing six weeks of play. Despite missing 18 games over that span, Horvat managed to set a new career-high in goals with 22, often playing on the first line with Bärtschi and rookie Brock Boeser.

On April 12, 2018, Horvat was named to Team Canada’s senior team to compete at the 2018 IIHF World Championship.

Travis Konecny, a forward for the Philadelphia Flyers, is Horvat’s second cousin. On September 3, 2018, he got engaged to his longtime girlfriend Holly Donaldson. On July 19, 2019, Horvat married Holly in a ceremony with several NHL players, including Jayson Megna, Josh Anderson and Max Domi, in attendance.

On October 9, 2019, Horvat was named the 14th captain of the franchise in a pre-game ceremony during the home opener of the Canucks’ 50th season. On October 20, during a game against the New York Rangers, Horvat scored his 100th career goal, helping the Canucks win 3–2. Two days later in Vancouver’s next game, Horvat notched his first career hat trick, leading the Canucks to a 5–2 win over the Detroit Red Wings after the team trailed in the game 2–0. After a months-long league-wide season suspension due to the COVID-19 pandemic, Horvat joined the Canucks in their return to play on August 1, 2020, as the team qualified for the 2020 Stanley Cup playoffs. The first-year captain would notch 10 goals and 2 assists for 12 points in 17 games, as the Canucks would win two series en route to a 7-game series elimination in Round 2 by the Vegas Golden Knights. Horvat’s 10 goals would be good enough for the 3rd highest goal scoring output in a playoff year by a Canuck, trailing only Pavel Bure and Trevor Linden’s respective 16 and 12 goal efforts in the 1994 Stanley Cup playoffs.