Brebbia graduated from Wellington High School in Wellington, Florida. He attended Elon University in Elon, North Carolina, where he was a pitcher for the Elon Phoenix baseball team. During his freshman year, Brebbia helped the Phoenix secure an at-large bid to the 2009 NCAA Division I Baseball Tournament. The Phoenix ultimately finished with a final record of 41-18 as they garnered the program's first top-25 finish since moving to Division I. Baseball America ranked Elon 24th in its final poll of the season.
The New York Yankees selected him in the 30th round of the 2011 Major League Baseball draft.  The Yankees released him in December 2013. Brebbia played the next two seasons in the American Association of Independent Professional Baseball before signing with the Arizona Diamondbacks in September 2015. The Cardinals selected Brebbia from the Diamondbacks in the minor league phase of the Rule 5 draft at the 2015 Winter Meetings. 
He made his major league debut in Colorado on May 28 along with teammate Paul DeJong. He remained with the team for the rest of the season, and he finished his 2017 rookie campaign with a 2.44 ERA, 51 strikeouts in 51.2 innings, and a 0.93 WHIP.