The Jazz (4-4) were made to work harder than expected for a first road win of the season, riding Millsap's 34 points along with 24 points and 17 rebounds from Jefferson to emerge victorious in the marathon encounter.
"We finally picked it up and pushed it into overtime," Jefferson told reporters. "By that time, it's time to pick up our game and come out with a victory. Unfortunately, it took three overtimes to do it, but we did it."
Trailing by 11 midway through the fourth quarter, Utah rallied and eventually tied the game at 104-104 on Jefferson's unlikely three-pointer with two seconds remaining.
The 6ft-10in (2.08m) Jefferson tossed up a shot that bounced around the rim and finally found the net for just his second career basket from beyond the arc.
Jefferson tied the scores again with just 19 seconds left in the opening overtime, forcing another extra session, in which John Lucas's late three-pointer staved off defeat for the home team
The visitors finally broke free in the third overtime session with a 9-1 run that was highlighted by a three-pointer from Millsap, who had seven points in the third extra session.
DeMar DeRozan led Toronto with a career-high 37 points, while Jose Calderón had 20 points and 17 assists but could not prevent the Raptors (1-6) from slipping to a fourth straight loss.
Toronto was without injured players Kyle Lowry (right ankle) and Landry Fields (right wrist), who both missed their second straight game.
Utah's Mo Williams returned from a one-game absence due to a strained hip and he made a solid contribution with 17 points and 14 assists.
Randy Foye added 20 off the bench as the Jazz avoided another defeat away from home in the opener of a four-game road trip for the team.
"Everybody just played their heart out and we told each other that we didn't want to lose ... we wanted this game even if it took six overtimes," Foye said.
The Miami Heat needed just about all of LeBron James' season-high 38 points as they edged out the Houston Rockets 113-110.
The Heat scored the final seven points of the game to snatch the win, with James accounting for five of them.
"It's the zone you hear about," James said afterwards.
"Certain players talk about getting into the zone. I wish I could get into it more. But when you're in it, you know how it feels, you know everything you put up and how you're going is pretty good."
Chris Bosh added 24 points and 10 rebounds while Dwyane Wade had 19 points.
Chandler Parsons had 25 points and Omer Asik 19 for Houston, both career highs.
Jeremy Lin, however, endured another humbling moment against the Heat.
His headline-making run with the New York Knicks last season was derailed against Miami last season when he went 1 for 11 from the field and coughed up eight turnovers.
This time out, he shot an airball from 3-point range with 10 seconds left, which was caught by Wade.
"Obviously, it's my responsibility to hit that shot," Lin said.
"It was a good shot for me, a quality shot and it didn't go in for me, unfortunately, in a crucial play of the game."
Miami's old rivals Boston collected another win too as Rajon Rando led them to 101-95 victory over the Chicago Bulls.
Rondo had 20 points, nine rebounds and 10 assists while Kevin Garnett added 15 points.
Luol Deng led the Bulls with 26 points and 11 rebounds.
Russell Westbrook led a fourth-quarter comeback as the Oklahoma City Thunder edged out the winless Detroit Pistons 92-90.
The Thunder reeled off 13 straight to turn a double-digit deficit into a 75-73 lead, and Westbook finished with 33 points.
Brandon Jennings also collected 33 points as the Milwaukee Bucks beat the Philadelphia 76ers 105-96, while Nikola Pekovic's 20 points were enough to help the Minnesota Timberwolves to a 90-82 win over the Dallas Mavericks, even though he left before the end of the game after joining Minnesota's lengthy injury list.
Goran Dragic scored 21 points to lead the Phoenix Suns to a 110-100 win over Denver, while Josh Smith's 19 points and 11 rebounds helped the Atlanta Hawks top the Portland Trail Blazers 95-87.
Meanwhile Philadelphia 76ers centre Andrew Bynum is expected to be sidelined until early December because of a bruised knee bone and could play his first game the following month, his team has confirmed.
The 25-year-old All-Star, who has battled knee injuries for much of his career, has yet to take the court for the 76ers since he was acquired from the Los Angeles Lakers in a 12-player, four-team trade in August.
Follow us @SprtNationalUAE