James scored 14 of his 41 points in the fourth quarter when the Cavaliers, who had trailed 84-75, rallied to win for the eighth time in 10 games.
"We played another great game," James said.
"We've been playing extremely well the last two months and once again we continue to play well, we continue to defend and that's what gives us a chance to win."
Bryant had 33 points for the Lakers, who lost for a third consecutive game.
Los Angeles had the ball for the final nine seconds but could not get a shot off until Bryant, after a pass from Luke Walton, launched a three-point attempt after the buzzer.
"I fumbled the ball a little bit, wasn't aware the clock was as low as it was and as soon as I passed it back to Kobe, I realised that that was a mistake and unfortunately it cost us big," Walton said.
James said Cleveland's defence in the final seconds was instrumental in the victory.
"For us to play the defence that we did and not even give them a shot was great by everybody that was on the court," James said.
Bryant said he was unhappy with the chances the Lakers had given away in recent games.
"We just have stretches of mental lapses, putting teams on the free throw line when you have a good cushion, giving them these opportunities to get back in the game," he said.
Bryant had 12 rebounds and six assists to go with his 33 points. Lamar Odom was the only other Laker in double figures, taking 14 points.
Zydrunas Ilgaiskas added 16 points and 11 rebounds for the Cavaliers.
The game was delayed for 12 minutes in the first quarter after raindrops began falling from the roof.
Hedo Turkoglu's game-ending three-pointer led the Orlando Magic to a 96-93 home win over the NBA-leading Boston Celtics, who lost for the fifth time in 10 games.
Turkoglu finished with 27 points and Dwight Howard added 18 points, 16 rebounds and five blocked shots.
Paul Pierce had 24 points for the Celtics, playing without Kevin Garnett who is sidelined with an abdominal strain.
Brandon Roy had 24 points, including the game's final five and the Portland Trail Blazers rallied from a 19-point deficit to defeat the Atlanta Hawks 94-93.
Dirk Nowitzki's 32 points, including 16 in the final quarter, helped the Dallas Mavericks hold off the Denver Nuggets 90-85 to win their seventh successive home game.
Powered by Al Jefferson's 40 points and 19 rebounds, the Minnesota Timberwolves handed the New Jersey Nets a ninth consecutive loss, 98-95.
The Utah Jazz beat the Houston Rockets 97-89, with Houston's Yao Ming missing due to an upper respiratory infection.
Leandro Barbosa scored eight consecutive points to spark a 12-0 fourth-quarter run in the Phoenix Suns's 88-77 win over the Chicago Bulls. Amare Stoudemire led the Suns with 24 points. Barbosa finished with 13. Kirk Hinrich had 31 for Chicago.
Mo Williams scored 25 points as the Milwaukee Bucks beat the Washington Wizards 105-102 in overtime, despite Caron Butler's 40 points for Washington.
The Seattle SuperSonics lost their 14th consecutive game, falling 103-101 to the Sacramento Kings on Kevin Martin's buzzer-beater.
Andris Biedrins grabbed 26 rebounds, the most in the NBA this season, and Stephen Jackson scored 25 points in the Golden State Warriors's 106-104 win over the New York Knicks.