"I know what it's like when you are a young team that was up 16 and then you look up and you're down one. That rim sure gets tighter," Randolph said.
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 on Jan. 14. They also 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.
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.
"We know how important each of these games are for us. We stuck with it and in the fourth quarter, we got stops and played together," Gasol said.
The Grizzlies, who kicked off a five-game road trip, are 15-4 in their last 19 against fellow West teams and completed their first season sweep of the Jazz in nine years.
NOTES: Grizzlies guard Tony Allen missed Monday's game with a stomach illness but played in this game, going scoreless in 10 minutes. The Jazz were without guard Alec Burks (ankle). ... The Jazz almost made it through the entire third quarter without committing a foul, but gave up a three-point play to Gasol with 14 seconds remaining. ... Hayward lost his shoe as he was fouled going to the hoop in the fourth quarter.