The Cavaliers tied the 1995-96 Grizzlies, then in Vancouver, and the 1997-98 Denver Nuggets. They can equal their own overall mark for futility with a loss in Saturday night's game at home against the Portland Trail Blazers.
Cleveland lost 24 straight games over the 1981-82 and 1982-83 seasons.
Zach Randolph had 29 points and 13 rebounds, Rudy Gay added 26 points and 11 boards, and the Grizzlies won their season-best fifth straight.
Mike Conley added 21 points and eight assists for Memphis, while Sam Young finished with 19 points, matching his season best.
J.J. Hickson matched his career high with 31 points and grabbed 15 rebounds. Ramon Sessions finished with 20 points, and Antawn Jamison scored 19 for Cleveland, which hasn't won since a Dec. 18 overtime victory over the New York Knicks.
Cleveland gave Memphis all it could handle through the first three quarters, but the Grizzlies put together 13 straight points to open the fourth en route to their eighth win in nine games.
The victory, coupled with Portland's 100-87 loss to Indiana, moved the Grizzlies into the eighth playoff spot in the Western Conference by a half-game.
The Cavaliers were without guard Daniel Gibson, who left the team earlier in the day and returned to Cleveland for personal reasons. Add to that Mo Williams, the second-leading scorer, and Anderson Varejao out with injuries, and Cleveland was at a personnel disadvantage. The 6-foot-9 Hickson started at center, and rookie Christian Eyenga, who was in the D-League a little over a month ago, was at forward.
Memphis hit its first 10 shots for an early eight-point lead. But Cleveland erased the deficit before the end of the first quarter with a 16-4 rally. Memphis already had eight of its 17 turnovers on the night at that point.
Cleveland took advantage of Memphis' lackadaisical approach and led by 10 before carrying a 59-52 lead into the break. Hickson already had 21 points, including 14 in the second quarter.
The Cavs led much of the third, and Jamison's inside basket with a second left in the period gave Cleveland an 84-83 lead heading into the fourth.
Notes: Randolph was chosen the Western Conference's player of the month for January before the game. Cavaliers coach Byron Scott was a member of the 1995-96 Grizzlies. One of Cleveland's eight wins was Nov. 27 over the Grizzlies in Cleveland. Memphis had not won five straight since the end of the 2005-06 season. The teams are among those that have played the most road games in the league. Memphis has 28 games away from home; Cleveland played its league-leading 29th on Friday. Cleveland plays nine of its next 10 at home.