Hagoromo's chakra allowed Sasuke to awaken the Rinnegan. Nothing else could have given Sasuke the Rinnegan, Hashirama's cells had no Ashura chakra since it leaves after death and Hagoromo said Madara got the Rinnegan by getting Hashirama cells before the transmigration was over. If Hashirama's cells had Ashura chakra then Sasuke would have 2 Rinnegans exactly like Madara. Obito said Indra got the Sage's eyes that gifted him with powerful chakra and spiritual energy. Then Obito tells Konan spiritual energy was Yin energy and Sasuke got the Yin chakra of the Sage's chakra.
Hagoromo gave them the seal to use CT on the Juubi Jin and gave them his chakra (Yin to Sasuke and Yang to Naruto). That chakra from the Sage gave both Naruto and Sasuke abilities such as Naruto's TSB/flight/healing and Sasuke's Rinnegan jutsus.