Cozart attended Collierville High School in Collierville, Tennessee, graduating in 2004. He played football, basketball, and baseball. In his senior year, he batted .443 with five home runs and 41 RBIs.

He attended the University of Mississippi, where he played college baseball for the Ole Miss Rebels. Baseball America named him a third-team All-American in 2006. Cozart had a .310/.367/.473 career slash line in college.

He returned to Dayton in 2008, this time hitting .280/.330/.457 for a full season. He hit 14 home runs, including two multi-homer games and one grand slam. Cozart drove in 49 runs and hit six triples in 418 at bats. His performance earned him a trip to the Midwest League All-Star game, and he was named a Baseball America Class-A All-Star. Midwest League managers voted him the best defensive shortstop of the year. Cozart led the Dragons in batting average for the season.

The Reds invited Cozart to spring training in 2010. He spent the season with the Louisville Bats of the Class AAA International League, hitting .255/.310/.416 in 553 at bats (first in the International League) with 91 runs (leading the league), four triples and 30 stolen bases (tied for 5th in the league; the fourth-most by any Reds minor league player). Cozart put up career highs in power with 17 home runs and 67 RBIs. He was named a Baseball America Triple-A All-Star for 2010, and received the Mary E. Barney Award as Louisville’s Most Valuable Player. As one of six players to spend all of the season with Louisville, he was fourth in the International League in games (136) and fifth in hits (141). He also led all IL shortstops with a .977 fielding percentage. Cozart was rated the 10th-best prospect in the Reds organization. On November 19 he was added to the Reds’ 40-man roster.

He was placed on the Reds 40-man roster after the 2010 season, but was optioned to Louisville on March 21, 2011. Cozart enjoyed a hot start to 2011, hitting .310/.357/.467 through his first 77 games. He went 100-for-323 with 26 doubles, two triples, seven homers, and 32 RBIs, along with nine steals. The Reds shortstops, Paul Janish and Édgar Rentería, were hitting a combined .226 with one homer and 29 RBI through July 6, and Reds fans everywhere were calling for a change at shortstop. On July 7, 2011, less than 24 hours after an article on Reds.com said there would be no transaction, the Reds optioned Janish and recalled Cozart. Switching uniform numbers from #60 to #2, Cozart started that night at Milwaukee, playing shortstop and batting seventh. In his debut game, he went 1-3 with a run scored. Cozart hit his first home run in the eighth inning off of his college teammate Lance Lynn, on July 17, against the rival Cardinals. He had 10 hits in his first 25 Major League at bats, and at least one hit in each of his first 6 games.

Cozart was named the opening day starting shortstop for the Reds on April 5, 2012. The Reds won that game 4-0 with Cozart going 2-4. On April 7, Cozart went 3-4 with a single, triple, and a home run going a double shy of the cycle in the Reds 8-3 loss. In 138 games, he finished the year batting .246/.288/.399 with 33 doubles, 15 home runs, and 35 RBIs.

In 2013, Cozart batted .254/.284/.381 with 30 doubles, 12 home runs, and 63 RBIs in 151 games. He also led the National League with 10 sacrifice flies.

Cozart had a down year at the plate in 2014, batting only .221/.268/.300, with both his on base percentage and his slugging percentage the lowest of all qualified MLB batters, and posting lower offensive numbers in nearly every statistical category. On October 23, 2014, he was nominated for the Gold Glove for National League Shortstops. In 147 games, he had 4 home runs and 38 RBIs.

In 2015, Cozart got off to a solid start. On June 10, during a game against the Philadelphia Phillies, Cozart slipped on first base, tearing the anterior cruciate and lateral collateral ligaments and the biceps tendon in his right knee. He would need reconstructive surgery following the injury, causing him to miss the remainder of the season. He batted .258/.310/.459 with 9 home runs in 53 games.

On April 4, 2016, Cozart batted leadoff on opening day for the Reds and went 3-for-3 with 2 doubles and an RBI in a 6-2 win over the Phillies. He finished 2016 batting .252/.308/.425 while hitting a career-high 16 home runs and 50 RBIs in 121 games. Despite all that, Cozart was hampered by the complications from the surgery on his knee in 2015, which eventually caused patellar tendinitis in his right knee and caused him to miss the final three weeks of the season.

Cozart had another career year with the Reds in 2017, batting .297/.385/.548 and setting career highs in home runs (24), triples (7; tied for 5th in the NL), and runs scored (80), while tying a career high with 63 RBIs in 122 games. He ranked among overall NL leaders in slugging percentage (10th), average vs LHP (.337, 6th), and hardest to GIDP (87.6 AB, 10th). He was voted to his first-ever All-Star game as a player and a starter for the NL. However, he played through most of the season with a quad injury, which caused him to serve two stints on the disabled list. He became a free agent after the season.

On December 15, 2017, Cozart signed a three-year, $38 million contract with the Los Angeles Angels.

On June 13, 2018, Cozart left a game with a shoulder subluxation while diving for a ball. Two weeks after his injury, it was revealed that he had suffered a torn labrum in his left shoulder, and he underwent season-ending surgery. Cozart finished his first season with the Angels batting .219/.296/.362 with 5 home runs and 19 RBIs in 58 games. He played third base, second base, and shortstop.

On April 25, 2019, Cozart was placed on the disabled list due to a neck strain. He was hitting .109 through the first month of the season, playing primarily third base. On July 12, it was announced that he would require surgery on his injured left shoulder and would miss the remainder of the season.

On December 10, 2019, Cozart and Will Wilson were traded to the San Francisco Giants in exchange for a player to be named later or cash considerations. He was released by the Giants on January 15, 2020.