Born in Sátão, Viseu District, Rodrigues joined S.L. Benfica's youth system at the age of 12 from local Associação Social Cultural e Recreativa e Desportista da Casa do Benfica de Viseu. On 26 August 2015, still a junior, he made his professional debut with the former's reserves, playing the entire 1–0 home win against U.D. Oliveirense in the Segunda Liga.[2]
Rodrigues scored his first goal in the competition on 13 February 2016, in a 2–2 away draw with Académico Viseu FC.[3] In the 2016–17 season, he netted four times (all from penalties) from 38 appearances to help the side finish in fourth position.[4]
Rodrigues scored his first goal in the Portuguese top division on 30 January 2018, helping the hosts defeat C.D. Tondela 3–0 with an early long-range shot.[8]
On 7 September 2020, Olympiacos F.C. announced the signing of Rodrigues on a five-year deal, for a transfer fee in the range of €4 million.[13] In the following transfer window, he was loaned to F.C. Famalicão until 30 June;[14] the move was extended for the entire 2021–22.[15]
On 21 August 2023, still owned by Olympiacos, Rodrigues joined Cypriot First Division club Pafos FC for one year.[18] In July 2024, he agreed to a permanent contract.[19]
^Serrano, Tiago (23 March 2017). "Olheiro BnR – Pedro Rodrigues" [BnR scout – Pedro Rodrigues] (in Portuguese). Bola Na Rede. Retrieved 20 February 2020.
^"Pêpê veste as cores estorilistas" [Pêpê wears Estoril colours] (in Portuguese). G.D. Estoril Praia. 21 August 2017. Retrieved 24 September 2017.
^"Pedro Rodrigues já fala "à Estoril"" [Pedro Rodrigues already talking "like an Estoril man"] (in Portuguese). S.L. Benfica. 22 August 2017. Retrieved 24 September 2017.
^Tempelis, Vassilis (7 September 2020). "Ολυμπιακός: Επίσημα ερυθρόλευκος ο Πέπε" [Olympiacos: Pêpê officially red-and-white] (in Greek). Sport 24. Retrieved 10 October 2020.