A native of Blaine, Minnesota, Bjugstad was named Minnesota's Mr. Hockey as a high school senior in 2010, then scored 98 points (54 goals, 44 assists) in 109 games in three seasons at the University of Minnesota, which won Western Collegiate Hockey Association titles in 2012 and 2013.

The nephew of Scott Bjugstad, who played 317 NHL games for the Minnesota North Stars, Pittsburgh Penguins and Los Angeles Kings, Bjugstad was selected by the Florida Panthers with the No. 19 pick of the 2010 NHL Draft.

He made his NHL debut April 6, 2013, against the Washington Capitals, and scored his first NHL goal April 27, against the Tampa Bay Lightning.

Bjugstad led the Panthers in scoring with 38 points (16 goals, 22 assists) in 76 games as a rookie in 2013-14 and finished 13th in voting for the Calder Trophy, awarded to the NHL's top rookie.

The 6-foot-6 center scored an NHL career-high 24 goals in 2014-15 and had an NHL career-high 49 points (19 goals, 30 assists) in 82 games in 2017-18.

Bjugstad was acquired by the Penguins in a trade with the Panthers on Feb. 1, 2019. He played 10 games in 2019-20 before sustaining a lower-body injury Nov. 15 against the New Jersey Devils. He had core muscle surgery and missed 46 games, returning March 5 against the Buffalo Sabres.

After playing for the Minnesota Wild and Arizona Coyotes, Bjugstad was traded to the Edmonton Oilers on March 2, 2023, for minor-league defenseman Michael Kesselring and a third-round pick in the 2023 NHL Draft. His Oilers debut was his 600th NHL game, a 7-5 loss to the Winnipeg Jets on March 4.

Bjugstad returned to the Coyotes when he signed a two-year contract July 1, 2023. He played the 2024-25 season for the Utah Hockey Club and signed a two-year, $3.5 million contract with the St. Louis Blues on July 1, 2025. Thirty-five games into his Blues tenure, Bjugstad was traded to the New Jersey Devils on Feb. 4, 2026.