[1] Kleber made his professional debut in Germany's top-flight Basketball Bundesliga during the 2011–12 season, representing the s.Oliver Baskets, a team based in his hometown Würzburg.

[7] On 1 July 2015, Kleber returned to Germany and signed a two-year deal with Bayern Munich.

[10] Kleber moved from end of bench piece to key rotation player in his first few years, increasing his points, rebounds, assists and blocks per game numbers in each of his first three seasons.

On 18 April 2022, during game 2 of the first round of the playoffs, Kleber scored 25 points and hit eight 3-pointers in a 110–104 win over the Utah Jazz.

[16] On 17 March 2023, Kleber made a buzzer-beating, game-winning three-pointer in a 111–110 win over the Los Angeles Lakers.

[17] Kleber helped Dallas reach the 2024 NBA Finals where the Mavericks lost to the Boston Celtics in five games.

[18] On 2 February 2025, Kleber, alongside Luka Dončić and Markieff Morris was traded to the Los Angeles Lakers in exchange for Max Christie, Anthony Davis and a 2029 first-round pick.