Rzatkowski was born in Bochum, West Germany. In his early youth, Rzatkowski played for the Bochum east district team SV Langendreer 04. In 1998, he started playing for the VfL Bochum youth team until 2009. Rzatkowski then signed a contract for VfL Bochum II in 2009. Only one year later he signed his first contract as a professional with the VfL Bochum. His first game as a professional player was on the seventh match day against SpVgg Greuther Fürth. He came into the game as a substitute for Roman Prokoph. In his second match he made his first appearance in the starting squad and scored the third goal in a 3–0 victory against SC Paderborn 07 during the home match on 26 November 2010. He also played against Union Berlin and VfL Osnabrück in his first season with the first team.
For the 2011/12 season Rzatkowski was sent on loan to Arminia Bielefeld. On 27 August 2011 Rzatkowski scored his first goal for Arminia Bielefeld in a 2–2 draw with SSV Jahn Regensburg. On 15 October he scored the loan goal for Bielefeld in a 1–0 victory over Kickers Offenbach. In his one year at the club Rzatkowski made 37 league appearances scoring 4 goals and providing 8 assists. He also helped the club capture the Westphalia Cup, scoring 4 goals in five matches during the competition. By reaching the final, they also qualified for the 2012–13 DFB-Pokal.
Return to Bochum
After a successful season with Bielefeld, Rzatkowski returned to Bochum. He scored his first goal back with the club on 30 October 2012 in a 3–1 victory over TSV Havelse in a DFB Pokal match. On 18 November 2012 Rzatkowski scored his first two league goals of the season for Bochum in a 5–2 victory over SV Sandhausen. He ended his season appearing in 25 league matches scoring 3 goals and providing 11 assists.
St. Pauli
Rzatkowski signed a three-year contract with FC St. Pauli in January 2013. He joined the club in the summer transfer window, becoming a key player for the team. On 13 September 2014 Rzatkowski scored his first goal for St. Pauli, the eventual game winner, in a 2–1 victory over FSV Frankfurt. On 5 April 2014 Rzatkowski scored a late winner in a 3–2 victory over SV Sandhausen. On 12 February 2016 Rzatkowski scored the lone goal for St. Pauli in a 1–0 victory over RB Leipzig. The following match day, 19 February, he also scored in a 3–1 loss to FSV Frankfurt. He went on to score in a third consecutive league match on 28 February in a 2–0 victory over MSV Duisburg. At the conclusion of the 2015/16 season Rzatkowski was honored by Kicker as the best defensive midfielder in 2. Bundesliga.
RB Salzburg
On 10 June 2016, Rzatkowski joined Austrian Bundesliga club, FC Red Bull Salzburg on a four-year deal. On 21 September 2016, Rzatkowski scored his first goal for Salzburg in a 7–1 victory over SC Mannsdorf in the ÖFB-Cup. On 23 October 2016 he scored his first goal in the Austrian Bundesliga in a 5–1 victory over SKN St. Pölten. On 19 July 2017, Rzatkowski opened the scoring for Salzburg in a 3–0 victory over Hibernians F.C. in the second qualifying round of the UEFA Champions League.
On 31 January 2018 it was announced that Rzatkowski was joining the New York Red Bulls on loan for the 2018 season. On 22 February 2018 Rzatkowski made his debut for New York, coming on in the second half in a 1–1 draw against CD Olimpia in the first leg of the round of 16 match of the CONCACAF Champions League. On 13 March 2018, Rzatkowski scored the eventual game winning goal for New York in a 3–1 victory over Club Tijuana in the CONCACAF Champions League, helping the club advance to semifinals of the Champions League for the first time. On 14 July 2018, Rzatkowski scored his first two MLS goals in a 3–2 win over Sporting Kansas City. On 24 January 2019, Rzatkowski joined New York on a permanent basis.
