South African Rautenbach will leave the Exiles this summer after seven years at the Aviva Premiership club.
And the Exiles have confirmed that 28-year-old Griffiths will be part of their squad for next season.
Griffiths made more than 100 appearances for the Swansea-based Ospreys, helping them to three league title successes.
"Cai is very solid at the set-piece, which is crucial when signing a tighthead prop," London Irish rugby director Brian Smith said.
"He is a direct replacement for Faan Rautenbach, who is leaving London Irish after seven years' service."