Klimenta began his youth career in the United States, playing as a midfielder and forward. He also played for the Olympic Development Program (ODP) for Cal North.
Klimenta joined United Soccer League club Sacramento Republic FC in March 2014, after being selected during an open tryout. Klimenta played as a defender and has started most of the league games in the 2014, 2015 and 2016 season.[3]
In January 2018 it was announced that Klimenta had signed for LA Galaxy of Major League Soccer, ending his tenure in Sacramento as the team's all-time leader in minutes, starts, and appearances.[4] He was released by the club on June 14, 2018.
On 4 April 2019, Klimenta joined USL side Reno 1868.[6]
On 5 June 2021, Klimenta joined USL Championship club Oakland Roots SC.[7] After three seasons with the club, he was not re-signed for the 2024 season.[8]
In May 2016, Klimenta was called up for to the Montenegro national team for the first time for a training camp and friendly against Turkey on 29 May. On this day, Emrah made his senior debut. In the process, he became the first Sacramento Republic player to receive a senior call-up while with the club.[10][11]