Bryant scored 42 points, the third time in five games he has reached at least 40. He has had no fewer than 36 points in any of the eight games he has played in January and has produced 313 points to date in the month.
Cleveland scored the first nine points of the fourth period to draw within three and had a chance to tie the contest when Daniel Gibson missed a 3-pointer with 6:40 to play.
The Lakers eventually rebuilt their lead to nine and withstood a pair of 3-pointers from the Cavaliers in the final minute.
Los Angeles improved its home record to 8-1.