Crawford attended Catholic Central High School in Windsor, Ontario, where he was a three-sport athlete (football, basketball and track and field). In football, he played running back, outside linebacker and kick returner. He was voted first team all-conference (Windsor & Essex County Secondary School Athletic Association) during every one of his four prep seasons. As a junior, he was named the conference’s MVP and received the Royal Arcanum Outstanding Athlete Award.
He practiced track and field as a freshman and sophomore, winning gold medals both years in the shot put and discus in his conference and at the OFSAA championships. In basketball as a senior, he was named the MVP in his conference and received the Bill Rogin Award, for the best player in the Ontario secondary schools.
He was recruited by Boise State University and Michigan State University, but the academic curriculum from his Canadian high school didn't meet NCAA requirements, so he enrolled at Bakersfield College in 2008. After concentrating on just playing football and the defensive end position, he was named first-team all-conference as freshman in 2008. In 2009, he was honoured as a junior college first-team All-America and was named to both first-team all-state and conference. He also was awarded as the defensive's most valuable player for the Renegades. He registered 42 tackles (28 solo), including 14 tackles-for-loss and eight sacks.
Considered a four-star recruit by Rivals.com, he was ranked as one of the most talented JUCO prospects in the nation. He committed to Boise State University.
In his first season at Boise State University, as a junior in 2010, Crawford appeared in each of the Broncos 13 games as a back-up defensive end. He registered 32 tackles on the season (13 solo), including 13.5 tackles-for-loss, tied for most on the team, and 7.0 sacks, the second-most on the team.
In 2011, Crawford started in 12 games, and he recorded 44 total tackles (20 solo), including 13.5 tackles for loss and 11 sacks. He was named first team All-Mountain West Conference for his performance.
He was diagnosed with a heart murmur at the national scouting combine.
Crawford was selected in the third round (81st overall) by the Dallas Cowboys in the 2012 NFL Draft. On June 13, he signed a 4-year deal receiving a $575,252 signing bonus and a $390,000 base salary in 2012. His base salaries from 2013-15 were $480,000, $570,000 and $660,000.
As a rookie, he played the left defensive end position in defensive coordinator Rob Ryan's 3–4 defense. He was a reserve player and finished with 33 tackles (18 solo) and 5 quarterback pressures.
The next year, defensive coordinator Monte Kiffin was hired to change the defense to a 4-3 alignment and Crawford was expected to backup the strongside defensive end position and play both defensive tackle spots in passing situations. On July 21, 2013, he suffered a torn Achilles on the first full day of training camp and was placed on the injured reserve list.
Entering 2014, there were initial plans to evaluate him in the 3-technique defensive tackle spot vacated by free agent Jason Hatcher, but after the signing of Henry Melton he was kept at defensive end. During the regular season Melton struggled as he recovered from offseason surgery and worked through a groin strain, so Crawford was forced to play defensive tackle in week 3 against the New Orleans Saints and would earn the starter job. He finished with three sacks and 29 quarterback pressures.
In 2015, he was signed to a five-year, $45 million contract ($25.7 million guaranteed). He was limited with a torn rotator cuff in his right shoulder that required offseason surgery, suffered in the second game of the season against the Philadelphia Eagles. He still managed 44 tackles, 5 sacks (third on the team), 27 quarterback pressures (third on the team).
In 2016, he struggled at the beginning of the season playing defensive tackle. The production of Terrell McClain and rookie Maliek Collins, combined with the need the team had at defensive end, made the Cowboys decide to move him to left defensive end in the fourth quarter of the second game of the season against the Washington Redskins. He would remain the starter at left defensive end for the rest of the season. His best game came in week 8 against the Philadelphia Eagles, registering 5 quarterback hurries, 1.5 sacks, 2 tackles for loss and a fumble recovery. He posted 2 sacks against the Cleveland Browns. He had to be replaced against the Tampa Bay Buccaneers with David Irving, because he was playing through shoulder and hamstring injuries. He did not play in the last 2 games of the season, because he was being saved for the playoffs. He finished third on the team with 4.5 sacks.