Clucas, who also attracted some late interest from Wolverhampton Wanderers, departs the Swans having made 36 appearances – 27 of which were starts – in his 12 months at the club.
Rowett is delighted to have pulled off the late transfer, a year on from Swansea paying double the price to sign the versatile ace from Hull City.
"I think this is a great deal for us as a Football Club, especially when you take into account the sum of money he moved for just a year ago," he told the club's official website.
"Most importantly for me though, he has a great character, he is incredibly mobile and he is exceptionally good technically - he gives us a lot of different options and we are delighted to welcome him to the club."
Chief executive Tony Scholes added: "Sam is undoubtedly a Premier League calibre player and has been one of Gary Rowett's prime targets during the transfer window, so we are delighted to have managed to get the deal over the line."
Clucas played a direct part in four Premier League goals during the 2017-18 campaign.