Damm began his professional career with Estudiantes Tecos at age 18. He made his debut in the Liga MX on 25 March 2012 in a 0–4 loss against Monterrey under coach Héctor Hugo Eugui. After his first season with Tecos, they were relegated to the Second Division where he remained for several seasons.

Damm completed a permanent transfer to Pachuca on 30 June 2013. During his first season with his new club, he appeared in every game and scored the first goal of his career against Guadalajara on 20 October 2013. In his second season with the club, Pachuca made it to the final of the Clausura 2014 season. He played in all of Pachuca’s playoff matches, reaching the final against Leon, where Pachuca lost 3–4 on aggregate score.

Damm started off the 2014 Apertura season with an injury in the second match against Monterrey. After having missed 8 games he made a full comeback on October 1, 2014. Jürgen also participated in the 2014–15 CONCACAF Champions League with Pachuca scoring a goal against Municipal on 24 September 2014.

On 10 June 2015, Damm joined Tigres UANL in a reported $10 million deal. On 3 October 2015, he scored his first goal for Tigres in a 1–0 victory against Atlas.

On 18 March 2015, Damm received his first call up for Mexico under coach Miguel Herrera for the friendlies against Ecuador on March 28 and Paraguay on March 31. He made his senior debut on March 28, 2015 against Ecuador, coming in as a substitute in the 84th minute of the game.

On 17 November 2015, Damm scored his first international goal against Honduras in a 0–2 away win for Mexico.

In May 2018, he was named in Mexico’s preliminary 28-man squad for the 2018 World Cup in Russia. However, he did not make the final 23.

At Tigres, Damm was a part of four Mexican championship teams winning the Liga MX Apertura in consecutive years from 2015 to 2017, while adding the Liga MX Clausura title in 2019. The midfielder was a mainstay for Tigres totaling 172 appearances in all competitions. He completed his Tigres career with 13 goals and 23 assists. Damm also played in 22 CONCACAF Champions League matches and made four appearances in the Copa Libertadores when Tigres was invited to the competition in 2015 and reached the final against River Plate.

On 1 July 2020, Damm joined MLS side Atlanta United after they paid $50,000 in General Allocation Money to Houston Dynamo who had Discovery Rights over the Mexican player.