"Sometimes best-laid plans, mother luck turns a different direction," said Lindsey, the former San Antonio executive. "We feel very fortunate tonight."
Burke and Gobert are both expected to be introduced by the team Friday in Salt Lake City.
The Jazz were the beneficiaries of a run on big men to start the draft, which started with UNLV's Anthony Bennett going unexpectedly to Cleveland at No. 1 and knocking Kentucky center Nerlens Noel out of the top five.
All along, the Jazz had a plan if Burke fell into a range they felt they could afford.
"We got fortunate, I think with the selections that went before us that allowed Minnesota to make the decision to move off the pick," Lindsey said.
The Jazz swept in, trading their two first-round picks, 14 and 21, to the Minnesota Timberwolves for the rights to Burke, who the Timberwolves drafted at No. 9. The picks the Jazz traded turned into UCLA forward Shabazz Muhammad and Louisville center Gorgui Dieng.
"I'm definitely excited going into Utah," Burke said, "playing with a young team, the type of talent they have.
Later in the first round the Jazz traded the No. 46 pick, Virginia Tech's Erick Green, and cash to Denver for the rights to Gobert, who was picked 27th.
Also, the Jazz announced late Thursday night that they acquired the rights to the No. 47 overall pick, Brazilian guard Raul Neto (who has been playing in the Spanish ACB League), from the Atlanta Hawks for a 2015 second-round pick they are owed by the Brooklyn Nets.
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 draft gives the Jazz the exciting point guard they were looking for to play alongside a young core of players that includes Gordon Hayward, Derrick Favors and Enes Kanter.
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.
Lindsey said credit for Burke's arrival, in large part, went to coach Tyrone Corbin, who was in his ear, saying Burke's ability on the pick-and-roll, his shooting and leadership were qualities the Jazz needed.
As for whether he would be expected to start, Corbin was evasive, even though as of July 1 Burke will be the only point guard on the roster.
"One of the things that everybody will start doing," Corbin said, "they'll look at our roster and immediately think he will start. ... He has an opportunity to be one of the guys that starts the game for us."
The Jazz worked out 67 prospects in preparation for the draft, but 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. However, when the Detroit Pistons selected Kentavious Caldwell-Pope, Burke said he "got a little nervous ... because I didn't really work out for any teams after that."
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.
"We had some heartburn when we gave up both picks [in the trade for Burke]," Lindsey said, "because there were good players at 21, Rudy being one of them."
The GM described Gobert as "unique with his length" and the amount he cares.
"He's not in it because he's tall," Lindsey said. "He's in it because he competes."