Haula played the 2008–09 hockey season with Shattuck-Saint Mary's in Faribault, Minnesota. There he tallied 24 goals and 55 assists for 79 points in 49 games played. In March 2009, Haula committed to the University of Minnesota. He would be only the second player born outside of North America to play for the Gophers, following future Wild teammate Thomas Vanek of Austria. Three months later, on 27 June 2009, Haula was drafted by the Minnesota Wild in the seventh round, 182nd overall, at the 2009 NHL Entry Draft.
Before attending the University of Minnesota, however, Haula first played at the junior level in the United States Hockey League (USHL) with the Omaha Lancers for the 2009–10 season while he attended Daniel J. Gross Catholic High School. His 28 goals and 44 assists with a plus-minus rating of +36 helped lead the Lancers to the Western Division finals. In his sole USHL season, he was named to the All-Star and All-Rookie teams.
After leaving the USHL, Haula totaled 42 goals, 82 assists and 124 points in 114 games during three seasons of college ice hockey with the Gophers. He then decided to forgo his senior year and final season of eligibility to sign professionally with the Wild.
On 7 April 2013, the Minnesota Wild signed Haula to an entry-level contract. During the 2013–14 season on 29 November 2013, the Wild recalled Haula from the Iowa Wild of the American Hockey League (AHL) to make his NHL debut, where he scored first NHL point, an assist, and he was named the game's third star in a 3–1 loss to the Colorado Avalanche. On 9 December, Haula was reassigned to Iowa. In his return to Minnesota, on 18 January 2014, Haula scored his first career NHL goal against the Dallas Stars in the first period. During 2014 Stanley Cup playoffs, he scored four goals and three assists in 13 playoff games.
Following the 2014–15 NHL season Haula became a restricted free agent under the NHL Collective Bargaining Agreement. The Minnesota Wild made him a qualifying offer to retain his NHL rights and, on 5 July 2015, Haula filed for Salary Arbitration.
The 2015–16 season saw a breakout year for Haula, as he set NHL personal bests in goals, assists, and points, more than doubling his previous highs in all three categories. He centered Minnesota's third line with Nino Niederreiter and Jason Pominville, quickly becoming Minnesota's most effective line offensively as well as acting as a shutdown line against the league's best.
At the completion of his fourth season with the Wild in 2016–17, as a restricted free agent Haula was left exposed for the 2017 NHL Expansion Draft. After agreeing to a three-year, $8.25 million deal with the Vegas Golden Knights, Haula was Minnesota's required selection for the draft on June 21, 2017. In exchange for selecting Haula, the Knights received prospect Alex Tuch and sent a conditional third-round pick in 2018 to the Wild.