The winning streak is the second best in NBA history; surpassed only by the 33 victories they achieved Los Angeles Lakers in the 1971-72 seasons.
James was the great figure who had 25 points, 12 rebounds, 10 assists, three rebounds and made two blocks, which made it possible for the Heat comeback complete a comprehensive 27 points handicap who led in the second part.
The star of the Heat was the one who took care of the two shots to score when staff was four seconds left and secure the win after the Heat were 2:29 minutes earlier without getting any points and allowed the Cavaliers to approach partial 95-96.
But guard Wayne Ellington and forward CJMiles failed two-point attempts that would have allowed the Cavaliers get an advantage or have achieved the victory.
Along with James, guard Mario Chalmers confirmed he has found his best flick of the wrist and had 17 points, second top scorer of the Heat.
The power forward Chris Bosh, Dwyane Wade, and reserve forward Shane Battier scored 11 points each, while veteran guard Ray Allen had 10 points.
The Heat, who are still in the fight to reach the mark of the 1971-72 Lakers, this time they faced the party more likely they were to lose from 1 February, to allow the Cavaliers a lead of 27 points in the third period.
But James and the Heat were able to react and also completed the biggest comeback so far this season.
The Cavaliers (22-46), who remained the basis of low franchise, Kyrie Irving and the Australian rookie guard Dion Waiters, both injured, suffered a fourth straight loss.
Ellington was the leading scorer for the Cavaliers to get 20 points, power forward Tristan Thompson had 18 and guard Shaun Livingston added 14 points.
While the rookie power forward Tyler Zeller had a double-double of 12 points and 11 rebounds, which left him as the best player of the Cavaliers under the basket.