The Cavs went for a freshman from Canada to open the draft for the second straight year and hope Wiggins works out better than Anthony Bennett.
Bennett was injured last summer, came into the season out of shape and made no impact, one of the reasons the Cavs were back in this spot again.
But Wiggins seems a much more ready product after averaging a Kansas freshman-record 17.1 points. He might have ended up as the top pick anyway, but became the best option for the Cavs when Jayhawks teammate Joel Embiid suffered a stress fracture in his right foot shortly before the draft.
"A thousand thoughts are going through my head right now," Wiggins said. "It's a dream come true."
Wearing a black tuxedo jacket with a white floral pattern, the guard slipped on a maroon Cleveland hat, hugged his supporters and went on stage to shake hands with Commissioner Adam Silver, who was calling the first round for the first time since replacing David Stern.
Milwaukee followed with another freshman, Duke forward Jabari Parker, who on Wednesday disputed that he was out of shape for his workout with Cleveland and performed poorly. Some consider him the more NBA-ready player after an All-American season for the Blue Devils.
Embiid went third to Philadelphia, drawing loud cheers from the many red-and-blue dressed 76ers fans who made the trip to Barclays Center in Brooklyn, as well as from Philadelphia guard Michael Carter-Williams, last season's Rookie of the Year who was sitting on the arena floor level.
The 76ers had two top-10 picks and could afford to take a chance on Embiid, the big man who may have ended up the top prize despite his lone season at Kansas ending early because of a bad back. But once the foot injury popped up during workouts, leaving his NBA debut uncertain after surgery, the two teams at the top passed.
"He worked so hard," Wiggins said. "He didn't let nothing get to him. He always stayed motivated. So I'm just proud. It's a proud moment for Kansas.
Arizona forward Aaron Gordon went fourth to Orlando, which also had two picks in the lottery, followed by Australian guard Dante Exum to Utah.
Then came two teams not used to picking so high: The Boston Celtics took Oklahoma State guard Marcus Smart, with the rival Los Angeles Lakers following at No. 7 with Kentucky forward Julius Randle.
The draft squeezed itself into the free agency frenzy that kicked off Monday, when Carmelo Anthony informed the New York Knicks he would become a free agent. LeBron James did the same the next day in Miami, and moves around the league since then appeared to be made with the pursuit of them in mind.
If James is interested in returning to Cleveland, a new teammate would be glad to welcome him.
"I want to win," Wiggins said. "If he wants to win, we'd be good together."