Rondae Hollis-Jefferson had 16 points and 12 rebounds, and DeMarre Carroll scored 18 points for the Nets, who have two straight wins for the first time since Jan. 19-21.
Andrew Harrison, who missed Memphis' last nine games due to a right wrist injury, scored 19 points, and Brionte Webber had a career-high 15. The Grizzlies have lost 20 of 21.
Memphis was without Marc Gasol due to flu-like symptoms. The team has lost 15 straight games on the road since a Christmas Eve victory at Sacramento.
Brooklyn trailed 94-87 heading into the fourth quarter before turning it around against the NBA's second-worst team.
Carroll capped a string of three straight 3s to tie it at 97 with 8:45 remaining in the final period. After a three-second defensive technical free throw by Harrison gave the Grizzlies a 101-100 lead with 6:54 left, the Nets scored nine straight points highlighted by LeVert's dunk with 4:07 left to lead 109-101.
Crabbe increased Brooklyn's advantage 116-108 with 38 seconds left, but Memphis made it interesting.
Wayne Selden Jr., who scored 17 points of the bench, hit a 3, and JaMychal Green made a layup to make it 117-115 with 12 seconds left. Spencer Dinwiddie then hit a free throw to extend the Nets' lead to 118-115 with 11 seconds to go before Harrison missed a potential tying 3 with 0.4 seconds remaining.
