Though Haula is a native of Pori, Finland, his hockey upbringing had a major American influence.

His father Tomi coached American football in Turku and in 2008, at the age of 17, Erik arrived in the United States to attend Shattuck-St. Mary's School in Faribault, Minnesota, a school that has turned out elite players such as Sidney Crosby, Nathan MacKinnon and Jonathan Toews. Haula, a forward with a left-handed shot and good hands, quickly adjusted to the North American game.

Haula's performance at Shattuck-St. Mary's earned him a scholarship to the University of Minnesota and attention from NHL scouts. He was expected to be an early-round pick in the 2009 NHL Draft but slid to the seventh round, where the Minnesota Wild selected him at No. 182.

He starred for Omaha of the United States Hockey League in 2009-10, scoring 72 points (28 goals, 44 assists) in 56 games to earn USHL All-Rookie and All-Star honors. Haula followed that by averaging better than a point per game during three seasons at the University of Minnesota. After signing with the Wild on April 7, 2013, Haula played six regular-season and five playoff games with Houston of the American Hockey League in the 2012-13 season.

Haula got his first chance in the NHL early in his first full pro season when the Wild promoted him from Iowa of the AHL on Nov. 29, 2013. He had an assist in his first NHL game and was named the game's third star. The Wild sent Haula down Dec. 9 after six games, but he returned later that month and became a fixture in Minnesota's lineup.

Haula appeared in all 13 of Minnesota's 2014 Stanley Cup Playoff games, scoring four goals and seven points to help the Wild advanced to the second round for the second time in their history.

He played the next three seasons with the Wild as a bottom-six forward before being taken by the Vegas Golden Knights in the 2017 NHL Expansion Draft. Haula had NHL career bests in goals (29), assists (26) and points (55), helping the Golden Knights set numerous NHL records for first-year teams.

Haula was traded to the Carolina Hurricanes on June 26, 2019. He played 41 games for the Hurricanes before he was traded to the Florida Panthers on Feb. 24, 2020. He was acquired by the New Jersey Devils in a trade with the Boston Bruins for forward Pavel Zacha on July 13, 2022. He signed a three-year, $9.45 million contract to remain with the Devils on June 23, 2023, after getting 41 points (14 goals, 27 assists) in 80 regular-season games and six points (four goals, two assists) in 12 Stanley Cup Playoff games.

Haula represented Finland at the 4 Nations Face-Off and was traded to the Nashville Predators by the Devils on June 18, 2025, for defenseman Jeremy Hanzel and a fourth-round pick in the 2025 NHL Draft. He helped Team Finland win bronze at the Olympic Winter Games Milano Cortina 2026.