To complete the deal, the Jazz selected UCLA's Shabazz Muhammad with the 14th pick, and Gorgui Dieng from Louisville with the 21st.
Later in the first round the Jazz traded the No. 46 pick and cash to Denver for the rights to 7-foot-1 French center Rudy Gobert, who was picked 27th.
In the buildup to the draft, general manager Dennis Lindsey said he would be aggressive, and did not fall short of the vow.
Burke averaged 16.7 points and 5.7 assists in two years at Michigan, where he led the Wolverines to the NCAA championship game in April.
The Jazz worked out 67 prospects in preparation for the draft, although Burke was not one of them. The Columbus, Ohio, native was projected in some mock drafts to go as high as No. 2 to Orlando, but the top of the draft was thrown entirely when the Cleveland Cavaliers drafted UNLV's Anthony Bennett No. 1 and Kentucky center fell to New Orleans at No. 6. The Pelicans subsequently traded Noel to Philadelphia in a package that included 76ers All-Star Jrue Holliday.
The Jazz added highly regarded talent at two positions at which they were thin.
After a 43-39 season, point guards Mo Williams, Earl Watson and Jamaal Tinsley will all head into free agency at midnight Sunday. In the frontcourt, the Jazz are anchored by Enes Kanter and Derrick Favors, but lacked rotational players.
Gobert, from Saint Quentin, France, has a 7-foot-9 wingspan and, while raw, was projected by many to go higher in the first round. He turned 21 on Wednesday.