The rookie and the MVP went back and forth in the second half, with Mitchell and the Jazz moving on to face the top-seeded Houston Rockets in a series that starts Sunday.
The Jazz took their biggest lead when Gobert hit two free throws to make it 86-73 with 7:04 remaining.
Westbrook's 3-pointer capped an 18-6 run and drew Oklahoma City to 92-91 with 1:28 to play.
Favors hit a jumper for the Jazz but then the Thunder corralled five offensive rebounds in a frantic set of possessions where Oklahoma City missed five shots. After a review, Paul George faked Gobert into the air behind the three-point line and leaned in but didn't get the call and shot an airball.
Mitchell scampered down the court as the Thunder yelled at the officials and eventually made two free throws with 6.9 seconds left to clinch the game.
After struggling to score in the first two periods, the Jazz just gave the ball to Mitchell and set screen after screen to allow him to take advantage of Oklahoma City's switching defense.
In the third quarter, Mitchell scored 22 points to boost the Jazz into a double-digit lead. The rookie scored on electrifying drives to the basket, spinning shots in the lane and rainbow three-pointers.
The arena, awash in gold as the Jazz wore their yellow jerseys and every fan was given a gold T-shirt, roared as Mitchell held his arms aloft and beckoned for more.
Mitchell picked up his fifth foul with 10:01 to play but returned to finish the game midway through the fourth quarter.
Seven minutes into the game, Rubio began limping and left the game.
Facing a team without a true point guard, the Thunder extended their defense and disrupted Utah's sets. The Jazz were shooting from unfamiliar spots and connected on just 39 percent in the first half.
Oklahoma City led by as many as nine before the Jazz tied it at 41 at halftime.
Raptors wrap up series with win over Wizards 102-92
The Toronto Raptors overcame a halftime deficit in a road playoff game for the first time in 24 tries, using 24 points from Kyle Lowry and a tremendous effort from their finally whole second unit to beat the Washington Wizards 102-92 in Game 6 on Friday night and end their Eastern Conference first-round series.
Toronto trailed by as many as 12 points in the first quarter and was down 53-50 at the end of the second. The Raptors' first lead did not come until the third quarter, and they were back down by five points entering the final period.
But with star guards Lowry and DeMar DeRozan, who had a personal series-low 16 points, resting to begin the fourth, reserves led by point guard Fred VanVleet turned around the game. Bothered by a bad right shoulder, VanVleet had played a total of three minutes in the series until Friday, but he was just what Toronto needed in this game, with five points, four assists and four rebounds.
The home team had won each of the first five games, but the Raptors were finally able to break that road hex and close things out. The East's No. 1 seed will face LeBron James' Cleveland Cavaliers or the Indiana Pacers next.
This marks the Raptors' third consecutive trip to the second round of the playoffs; they last failed to get that far in 2015, when they were swept in the opening round by the Wizards.
But this time, Washington bows out earlier than it was hoping when the season began with grandiose talk of a run to the conference finals. Instead, the Wizards were without John Wall for 41 games and wound up with the No. 8 seeding.
Wall had 23 points and eight assists in Game 6. Bradley Beal led Washington with 32 points. But the Wizards' run of eight home playoff wins in a row came to a close.
Surprising that Toronto would do it, perhaps, given that it had been 0-23 when trailing at halftime in an away game in the postseason, the team said, citing the Elias Sports Bureau.
Even as DeRozan sat out two-thirds of the fourth quarter, the Raptors chipped away at Washington's lead.
The Raptors never led by more than one point until a little more than 8 1/2 minutes remained, when reserve C.J. Miles made a three to put the visitors ahead 84-80, shortly after a three by VanVleet had erased the last bit of Washington's last lead with a three of his own.
Pacers blowout Cavaliers 121-87 to force Game 7
Victor Oladipo scored 28 points and posted the first postseason triple-double of his career Friday night, leading the Indiana Pacers to a 121-87 blowout over the Cleveland Cavaliers to even the series at three.
Game 7 will be played Sunday in Cleveland.
The three-time defending Eastern Conference champs again were led by LeBron James with 22 points, seven assists and five rebounds despite sitting out the entire fourth quarter. Nobody else scored more than 13 points as the Cavaliers lost their first closeout game in 14 tries dating to 2009. James had won 11 consecutive closeout games and is 12-0 all-time in the first-round series.
With its season at stake, Indiana played hard, aggressive and tough.
And it showed with a record-breaking victory margin.
Kevin Love hurt his left wrist on a hard fall midway through the second quarter and wound up scoring just seven points while shooting 3 of 10 from the field.
James took a shot, too, cutting the side of his left eye when he ran into Thaddeus Young's elbow on a drive late in the first half. No foul was called and James returned with a bandage protecting his face.
But with a relaxed Oladipo figuring out how to defeat the Cavaliers' double teams, he rediscovered his shooting touch. Oladipo finished with 13 rebounds, 10 assists and was 11 of 19 from the field and 6 of 8 on threes to keep the Pacers in charge most of the game.
After the Pacers erased a 22-17 deficit with a 12-2 run, the Cavaliers never led again.
Indiana's 10-2 spurt midway through the second quarter made it 46-36. The Pacers opened the third by scoring the first five points to make it 62-47 and then they poured it on.
Myles Turner's three-pointer with 7:03 left in the third made it a 20-point game and the Pacers closed the quarter on a 10-1 run to extend the lead to 92-67.