Kozlov made his ATP World Tour debut as a wildcard in 2013 Hall of Fame Tennis Championships at the age of 15.

In 2014, Kozlov reached two junior Grand Slam finals, where he lost to Alexander Zverev at the Australian Open and Noah Rubin at Wimbledon.

In 2013, Kozlov made his ATP World Tour debut as a wildcard at the Newport tournament and lost in the first round.

At the age of 16, he defeated his first Top 100 player in Tim Smyczek to reach his first ATP Challenger final in 2014 in Sacramento, where he lost to Sam Querrey.

In June, he won his first career ATP level match as a wildcard at the 2016 Ricoh Open in Den Bosch.

In September, at the 2021 Columbus Challenger, Kozlov won his first title in four years, defeating Max Purcell in the final in three sets.

[9] As a qualifier at the 2022 Delray Beach Open, he reached his second career quarterfinal after defeating Emilio Gómez[10] and Steve Johnson.

[12][13] At the 2022 Wimbledon Championships, he made his debut at this Major[14] when he entered the main draw as a lucky loser after the late withdrawal of Borna Ćorić.