After all, the Memphis Grizzlies were in the same position a few years ago as a young team that didn't know how to finish games.
"We were at that stage not too long ago. We would play a good game for 36 minutes or more and then a veteran team would execute, get stops and take the win away from us," said Gasol, who had 18 points.
Conley scored 19 points for the Grizzlies, who won for the fifth time in six games and are a season-best 15 games over .500.
"When the game is coming down to the end as a young team, you tend to try and make the right play and over-think things. We could see that in them and we tried to take advantage," Conley said.
Though Memphis is the team in a tight playoff race, the Jazz made the extra effort for steals and dived for loose balls and led for most of the contest while playing their starters extended minutes.
"They played so hard and they had extra energy and they had us on our heels most of the game, but we felt like we were supposed to win," said Randolph, who had his 400th career double-double.
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.
"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 only 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 past 19 against fellow West teams and completed their first season sweep of the Jazz in nine years.
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.