Maseko had long been tipped for a move to Copenhagen, with club legend Sibusiso Zuma personally recommending the 22-year-old.
He becomes the fourth South African to play for the club after Zuma, Elrio van Heerden and Luther Singh.
"In Thapelo, we get an explosive, dynamic player with exceptional qualities in direct, one-on-one situations,” Copenhagen coach Bo Svensson said. “His pace and ability to challenge his direct opponent will make him a constant threat in behind, but he can also use his left foot to assist teammates when cutting inside.
"There are naturally still elements to refine, but Thapelo's potential is high, and he always works tirelessly for the team, so we are really looking forward to working with him on a daily basis."
Maseko ends a frustrating spell at Sundowns, where he initially excelled before being frozen out of the first team. He spent part of last season on loan in Cyprus.
"I am very proud of this agreement. It is a dream come true,” Maseko said. “To me, FC Copenhagen are a very special club. They are a club that is highly recognised internationally, and Zuma has told me so many great things about the club, the fans and the city. I am looking forward to experiencing it all myself.
"I am truly honoured to get this opportunity, and it fills me with hunger and ambition. I have come here to make a difference and leave my mark on the club. I have big ambitions for the club and on a personal level. I come here to win, and I am very excited to get out on the pitch and give everything I've got."