Mike Conley scored 19 points and Marc Gasol added 18 for the Grizzlies, who won for the fifth time in six games and are a season-best 15 games over .500.
Derrick Favors had 22 points and 10 rebounds, Gordon Hayward scored 18 points and Richard Jefferson added 15 for the Jazz, who are 1-5 in their last six home games.
Though Memphis is the team in a tight playoff race, the Jazz made the extra effort for steals and dove for loose balls and led for most the contest while playing their starters extended minutes.
The Grizzlies, however, made the key plays down the stretch and sank their final 13 free throws to secure the victory.
Jefferson made a 3-pointer and then added a fast-break layup to give Utah its largest lead, 65-49 midway through the third quarter.
The Grizzlies clawed back and by the time Mike Miller hit a rare 3-pointer for Memphis, the lead was down to 78-75 with 7:08 to play. Favors dunked to put the Jazz up 84-77, but the Grizzlies clamped down and went to work.
Miller hit another 3 in the midst of 12 straight Memphis points and Gasol's layup capped the run to put the Grizzlies up 89-84 with 59 seconds left in the game.
Memphis forced six fourth-quarter turnovers and outscored the gasping Jazz 29-16 in the final frame. Trey Burke, who had 11 points and 10 assists, cut the lead to 89-87 with 15 seconds left but couldn't convert on a 3-pointer in the closing moments to extend the game.
The Grizzlies have the third-best record in the NBA, 26-9, since Gasol returned from a left knee sprain. And they have the league's best defense in that stretch, allowing just 90.2 points per game.
As good as they defend in the half court, the chink in the Grizzlies' armor is transition defense. The opportunistic Jazz pushed the ball in the first half and the Jazz big men challenged their more accomplished counterparts by going strong to the basket and banging for rebound position.
Enes Kanter had 11 points and a season-high 15 rebounds, but couldn't make his jumpers when Gasol sagged off him.
Memphis was kicking off a five-game road trip, with visits to Golden State, Portland, Denver and Minnesota still to come.
The Grizzlies are among five teams, including Portland, Golden State, Dallas and Phoenix, who are within two games of each other and fighting for the final four playoff berths in the West with three weeks left in the regular season.
The Grizzlies are 15-4 in their last 19 against fellow West teams and completed their first season sweep of the Jazz in nine years
Copyright 2016 Nexstar Broadcasting, Inc. All rights reserved. This material may not be published, broadcast, rewritten, or redistributed.