The Northern Ireland youngster has agreed a deal with Brendon Rodger's side until June 2015, and has passed a medial with the Welsh club
The 19-year-old elected to sign a deal which keeps him at the Liberty Stadium until June 2015, despite interest from Everton and Liverpool.
The Northern Ireland Under-21 international passed a medical on Friday and is set to join his new side's training next week.
"We are delighted to be able to attract one of Northern Ireland's biggest talents, I am really looking forward to helping Rory fulfil his potential and, hopefully, over the next few years ensure that Swansea City has another exciting young player blossoming at the club,'' said Swansea manager Brendan Rodgers.
Donnelly has described the move as a 'dream' but thanked his former club for their commitment to fielding young players, which paved the way for a switch to the Premier League.
"Everything has happened so quickly that I still can't quite believe it, it's a dream come true," said Donnelly.
"I was really impressed with what the manager had to say when I met him. He couldn't have done anything more to impress me.
"I believe I am signing for a brilliant manager who is prepared to give youth a chance. I also met the likes of Scott Sinclair and Joe Allen when I was over here last week and it only reinforced the belief that Swansea is the right club for me.
"I know there was serious interest from Everton and Liverpool, but I love the way Swansea play the game and I think it will suit me.
"The manager has told me there is no pressure on me because I am only 19. In time there will be pressure, but not right now.
"Like Swansea, Cliftonville had faith in me and I'm grateful that manager Tommy Breslin was also prepared to give youth a chance.
"I'd like to thank the club and the fans, manager and the coaching staff for developing me as a player. I hope I've repaid them with goals and a transfer fee.
"But now I just can't wait to get started at Swansea. I've worked hard for the chance, but I'm still grateful and thankful to be here.''