The Australian, who left Catalans Dragons at the end of last season, has signed for the Robins on a one-year contract.
He played for St George-Illawarra Dragons and Wests Tigers in the NRL and has Super League experience with London Broncos and Leigh Centurions as well as Catalans.
Drinkwater, who kicked four goals when the French side beat Warrington Wolves at Wembley in August, said: “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, who is due to arrive on January 3, 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.”
Meanwhile, Huddersfield Giants’ 20-year-old scrum-half Oliver Russell has joined Leigh Centurions on a season-long loan.
Russell, son of former Wigan, Oldham and Castleford player Richard Russell, made six Super League appearances for Giants in 2018, kicking eight goals and a drop goal.