Gasol and Randolph each had 10 rebounds as Memphis won its eighth straight home game and fifth straight over the Jazz. Mike Conley finished with 18 points and seven assists, and Tony Allen scored 10 points.
Gordon Hayward, Trey Burke and Alec Burks each scored 16 points for Utah, while Enes Kanter had 12 points and 12 rebounds. The Jazz lost their sixth straight and 11th of their last 12.
Memphis weathered several runs by the Jazz before six straight points in the fourth quarter provided a sufficient buffer. The Grizzlies began the night in the eighth and final playoff spot in the West.
Utah kept making runs at Memphis and finally caught the Grizzlies near the midway point of the fourth.
Memphis had carried its advantage to 12 early in the period, but Utah answered with a 16-4 run to tie the score at 84 with 5:38 left. The tying basket came after Memphis' fourth turnover of the quarter, leading to a breakaway by Burks.
Helping the Utah run was the Jazz hitting 10 of their first 12 shots in the fourth.
But Memphis got inside for its next three baskets - by Tayshaun Prince, Randolph and Conley - to increase the lead to 90-84, enough for the Grizzlies to secure the game.
The Grizzlies outrebounded Utah 46-39 and outscored the Jazz in the paint 54-40. Utah offset Memphis' inside play going 7 of 18 on 3s, including 4 of 9 after halftime.
As poorly as the Jazz played in the first half when they trailed by as many as 18, they reversed their fortunes in the third quarter.
Hayward, who converted only one of eight shots in the first half, got untracked, hitting his first three shots of the second half and helping Utah go on a 16-4 rally to get the Jazz close.
At that point in the third, Memphis had managed only three field goals on its first nine shots.
Utah would get within 3, but Memphis responded to run the advantage back to 13 before holding a 74-63 lead heading into the final frame.
Hayward scored eight in the quarter, while Richard Jefferson had nine for Utah, which made 13 of 22 shots in the third.
Memphis shot 56 percent in the first quarter to build an early lead before holding a 48-32 advantage at halftime. The point guards led the early scoring for both teams. Burke had 13 at the half, while Conley scored 10, all in the first quarter.
Randolph had 11 points and eight rebounds for Memphis, which led by 18 late in the half.
Burke was 5 of 8 for Utah, but the rest of the Jazz were 8 of 32.
Memphis already held a 29-19 advantage on the boards at the break and had outscored Utah 28-12 in the paint.