Hazelbaker was selected by the Boston Red Sox in the fourth round of the 2009 MLB Draft out of Ball State University. He opened 2009 with the Lowell Spinners, before being promoted to the Greenville Drive, Salem Red Sox, Portland Sea Dogs, and Pawtucket Red Sox. He batted a combined.259/.345/.434 line with 49 home runs and 157 stolen bases in 409 games. Hazelbaker returned to Pawtucket in 2013. He posted a.257/.313/3.74 line with 11 home runs and 54 RBI in 121 games, including 62 runs scored, 37 stolen bases, and 26 extra bases.
On March 31, 2016, the Cardinals announced Hazelbaker had made the major league roster; he played primarily as a backup outfielder and pinch hitter. He made his major league debut on April 3 as a pinch hitter against the Pittsburgh Pirates after 751 games in the minor leagues. His hitting performances in the first two weeks of the season thus caught national headlines. Hazelbaker's first MLB hit was a double on April 5 and first home run was the next day; both hits were against the Pittsburgh Pirates. On April 8, Hazelbaker's second major league home run became part of a unique event in major league history. Playing against the Atlanta Braves, he, Aledmys Díaz, and Greg Garcia established a new MLB record by each hitting a pinch hit home run, becoming the first trio to hit three pinch hit home runs in one game for the same team. The previous record of two pinch hit home runs by the same team in the same game had previously occurred on 57 occasions, most recently in 2011. The Cardinals won this contest, 7–4. That home run also gave him two in his first four career games, joining Joe Cunningham as the only Cardinals to accomplish the feat. His first career four-hit game, including his first career triple, came in a 10–1 victory over the Milwaukee Brewers on April 11. In the month of April, he batted.317,.357 OBP,.683 SLG, five home runs, and 13 RBI. The Cardinals optioned Hazelbaker to Memphis on June 15, 2016, after hitting.250 with seven home runs and 19 RBI. During a game against the New Orleans Zephyrs on June 28, Hazelbaker was instantly ejected when he charged the mound for José Ureña, who had nearly hit him in the head. Hazelbaker also flipped his bat and avoided a left punch from Ureña. After playing 13 games for Memphis and batting.325 with one home run and 11 RBI, St. Louis recalled Hazelbaker on July 19, 2016.
Arizona Diamondbacks organization
On November 4, 2016, the Arizona Diamondbacks claimed Hazelbaker off waivers. He was designated for assignment on April 6, 2018.
On July 28, 2019, Hazelbaker signed with the Sioux City Explorers of the independent American Association. He played 35 games with Sioux City, slashing.248/.421/.774. Hazelbaker was released on March 13, 2020.
