Earlier this week, it was reported that Arsenal had edged ahead of Manchester United in the race to sign the 16-year-old.
Sunderland have now confirmed that the teenager has completed a switch to North London after financial terms were agreed between the two teams.
A statement read: "Sam declined the offer of scholarship terms with the club and made it clear that he saw his future elsewhere.
"He expressed his desire to join Arsenal and we therefore moved quickly to bring the matter to conclusion, believing a swift resolution to be in the best interest of both Sam and the club.
"Terms were agreed with Arsenal and everyone at Sunderland AFC wishes Sam the very best of luck for the future as he embarks on the next stage of his football career."
Greenwood is expected to feature for Arsenal's Under-18 squad during the upcoming campaign.