The reversal means that the NBA champions have lost three games in a row for the first time this season.
Bradley Beal scored 33 points as the Wizards claimed five wins in a row for the first time in three years.
Russell Westbrook, who had 32 points, plus 14 rebounds and nine assists, made a shot inside and was fouled with 11.6 seconds remaining in overtime to put the Wizards up by three points.
Although he missed the free throw which gave the Lakers the chance to tie the score, they were unable to make the shot.
James finished with 31 points, 13 assists and nine rebounds while Montrezl Harrell added 26 off the bench
Elsewhere, Donovan Mitchell finished with 23 points and eight assists and three of his Utah Jazz team-mates scored at least 20 points each off the bench as their side beat the Charlotte Hornets 132-110.
The Miami Heat defeated Oklahoma City Thunder 108-94 while Devin Booker hit 32 points as the Phoenix Suns were comfortable 132-100 winners over the Portland Trail Blazers
After an eight-day rest because of two weather-based postponements, the Dallas Mavericks beat the Memphis Grizzlies 102-92 while the Chicago Bulls were too strong for an under strength Houston Rockets, winning 120-100.