The 19-year-old becomes Derek McInnes’ seventh summer signing after captaining Partizan last season.
A product of Partizan’s youth academy, he spent time on loan at second-tier FK Teleoptik before returning to make his senior debut in the 2024/25 season.
"It feels amazing, and I'm really happy and proud to be part of such a big club like Rangers,” Dragojevic told club media.
"It's a big opportunity, and I can't wait to get started.
"I first heard (the interest) three months ago, and when I heard it I already knew it was very interesting to me because I think the winning mentality that Rangers have is compatible with my mentality."