The skid had been the worst for Denver since the Nuggets lost 10 in a row 11 seasons ago.
Lawson's 3-point shot came with 54 seconds to play and gave Denver a six-point lead. Memphis scored two quick baskets, however, to close within 110-108 and the Grizzlies then put Nate Robinson on the foul line with four seconds remaining.
Robinson hit his first attempt, but missed the second to keep Memphis alive. Mike Miller missed a 3-point try at the buzzer that would have brought about overtime.
Lawson had 18 points and 12 assists while Kenneth Faried and Timofey Mozgov each had 16 points.
Denver shot 53 percent from the field and managed the win despite making 13 fewer free throws than did Memphis.