The 26-year-old has played in Super League with London Broncos, Leigh Centurions and Catalans Dragons and kicked four goals when the French side beat Warrington Wolves at Wembley in August.
Drinkwater began his career with St George Illawarra Dragons in 2013 and had another stint in the NRL with Wests Tigers two years later.
He is currently in Australia, but due to arrive at Rovers on January 3.
Drinkwater said: “I can’t wait to get over there and get started with training with the season now pretty much only around the corner, which is pretty exciting.
“I had offers from other clubs, but Tim Sheens was a big contributing factor to me choosing Hull KR.
“Growing up in Australia he was the Australia head coach and after speaking to him on the phone about the club, I really felt like the club is building towards taking the next big step and I can’t wait to be a part of it.
“I’ve played at KCOM Craven Park before and when you play there you certainly know you are playing an away game, so I imagine it’s a great atmosphere to play in front of as the home team and I’m very excited to get out there and play in front of the fans.”
Sheens believes Drinkwater will be a key addition to Rovers’ squad.
“Josh will obviously add great depth to our halves which is important for us in what is a long season, especially with what happened last year with Danny McGuire being out for a fair amount of the season where we really struggled to replace him,” Sheens said.
“On the back of a great season for Catalans Josh brings us experience, a good kicking game, goal kicking and the depth to the squad which leaves me feeling comfortable that we have enough in the halves to get us through the season.
“He is a player we have been looking at and discussing for a while so I’m very happy to finally get it over the line.”