The Magic had a chance to win the game with less than 10 seconds to play, but Glen Davis' jumper was partially blocked and rebounded by the Bulls with just 1.5 seconds remaining.
The Bulls have now won the last seven matchups with the Magic played in Orlando and improved to 4-1 since trading Luol Deng.
The Magic extended their season-high losing streak to nine games.
Orlando played without Arron Afflalo and Nik Vucevic. Afflalo missed his third straight game with a strained right foot, and Vucevic was out for the fifth straight with a concussion.
The game featured 20 lead changes and 14 ties.
During their most recent road trip, the Magic were tripped up by injuries as they dropped five straight games.
Despite losing by an average of 20 points during the trip, coach Jacque Vaughn continually lauded his team's improvement throughout.
Orlando returned home still battered, but the strides showed up again against a Bulls team that came in with three wins in their first four games since trading Deng to Cleveland.
In a game that didn't see either team take more than an four-point lead in any of the overtime periods, the Magic went ahead 125-122 in the third OT on a 3-pointer by Harris.
But Chicago kept chipping away, and eventually took the lead for good, 126-125, on a jumper by Tony Snell.
The Bulls had a 119-116 lead in the second overtime on a tip-in by Noah.
After several empty possessions on both ends, Davis tied it on a straightway 3 with just 4.2 seconds left.
Noah got free on a drive on the other end, but was fouled.
The Bulls then failed to get up a shot.
The Magic controlled the early pace of first overtime and took a 111-108 lead on a free-throw by Harris with 1:22 remaining.
But the Bulls committed their 17th turnover of the night on a bad pass by Noah with less than a minute to play, prompting a Magic timeout.
Nelson briefly got free, but came up empty on a 13-foot jumper, giving it back to Chicago.
Coming out of a timeout Mike Dunleavy got free on the wing and sank a 3-pointer to tie it with 13.4 seconds left.
Nelson dribbled the clock down, but couldn't get a shot off before time expired.
The Magic hovered above 55 per cent shooting as they built as much as a 15-point lead in the third quarter.
NOTES: The Magic are now 0-11 without Vucevic this season. ... The Bulls recalled G Marquis Teague on Wednesday from the Iowa Energy of the NBA Development League.