Roberts has impressed on loan at Birmingham in the first half of the season and has now signed a two-and-a-half year contract for an undisclosed fee.
It brings to an end Roberts' four-year stay at Sunderland, with the former Manchester City youngster making 147 appearances for the Black Cats. He scored nine goals and provided 18 assists.
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Roberts scored a dramatic winner in the 2022 League One play-off semi-final victory over Sheffield Wednesday, which led to promotion to the Championship.
And he played through injury to come off the bench and help spark the turnaround in last season's play-off final victory over Sheffield United.
Speakman said: “We are really pleased for Patrick. His loan spell at Birmingham City has been incredibly successful, and he has been one of the standout performers in the league.
"He is settled, performing, and his family are happy, and knowing this makes it the perfect way to end his time at Sunderland.
"Patrick believed in Sunderland when we needed him, and we believed in Patrick when he needed a home. He has been an integral part of our story, and he will always be held in high esteem by our supporters. We wish him every success for the future.”
After completing his Birmingham move, Roberts said: "It feels great. It was one of those that I was eager to get done and I'm glad it's done now. We have had a good run of late, picked up some good results and it's time to kick on now for the rest of the season.
"To come to a club like this and start how we have started is great. I want to continue that and show the fans what I can do for the next few months and two years and hopefully get promoted.
"The signings we have brought in this January is going to really help us and bring competition to the squad and it's going to push us on in the league.
"It's all positive, there's good buzz around the club at the minute and I'm glad to have signed and be a permanent player."