The New York Yankees selected Burawa in the 12th round of the 2010 Major League Baseball Draft, and signed him for a signing bonus of $300,000. That year he pitched seven innings in relief for the Staten Island Yankees of the Class A- New York-Pennsylvania League. In 2011, he pitched for the Charleston RiverDogs in the Class A South Atlantic League, going 3–2 with 3 saves and a 3.63 ERA. He then pitched for the Tampa Yankees of the Class A+ Florida State League, going 2–2 with 2 saves and a 3.66 ERA. During spring training in 2012, Burawa suffered a tear to his oblique muscle and a cracked rib, which cost him the entire season. At the time his fastball was 93–95 mph, and he also threw a slurve at 75–78 mph. In 2013, he pitched for the Trenton Thunder of the Class AAEastern League. He pitched to a 6–3 win–loss record with 4 saves and a 2.59 earned run average and 66 strikeouts in 66 innings pitched over 46 relief appearances. He was throwing a 95–98 mph fastball, and a mid 80s slider. The Yankees invited Burawa to spring training in 2014. He pitched for the Trenton Thunder, going 0–0 with 1 save and a 1.59 ERA and 18 strikeouts in 17 innings in 11 relief appearances. He also pitched for the Scranton/Wilkes-Barre RailRiders of the Class AAAInternational League, going 3–1 with 3 saves and a 5.95 ERA and 55 strikeouts in 42.1 innings in 31 relief appearances. After the 2014 season, the Yankees added Burawa to their 40-man roster. Burawa began the 2015 season with Scranton/Wilkes-Barre. After pitching to a 1.75 ERA in 26 appearances, the Yankees promoted him to the major leagues on June 21. He made his major league debut that night, allowing four runs on three hits in of an inning, and was optioned to the minor leagues after the game. Burawa was designated for assignment on August 5. For the season, with Scranton/Wilkes-Barre he was 1–3 with 1 save and a 2.55 ERA in 32 games, and in 3 starts with Trenton he had a 3.27 ERA.
Atlanta Braves
On August 14, 2015, the Atlanta Braves claimed Burawa off of waivers, They assigned him to the Gwinnett Braves of the International League, where in 4 relief appearances he had a 2.08 ERA. He was called up by the Braves on September 1, when rosters expanded. He appeared in 12 games in relief for Atlanta, and had a 3.65 ERA. In 2016, he pitched for the Mississippi Braves of the AA Southern League, and was 2–0 with a 6.75 ERA in 9 relief appearances. Pitching for Gwinnett, he was 1–0 with 1 save and a 5.14 ERA in 7 relief appearances. He was released on June 3, 2016.
On April 6, 2017, he signed with the Bridgeport Bluefish of the Atlantic League of Professional Baseball. In 2017, he was 2–2 with a 7.78 ERA in 22 relief appearances. He was released on June 26, 2017.