The Lakers were able to stop an Augustin drive, deflecting the ball out of bounds with just 0.9 left on the clock -- but that was enough for Mike Dunleavy to inbound the ball to Gibson as he made his way to the basket.
The shot went off Gibson's fingertips with less than two-tenths of a second before the buzzer.
Young led all scorers with 31 points, including three free throws at the end of regulation to force overtime.
Augustin scored 27 for the Bulls on 10-for-16 shooting, including 5 for 7 from three-point range.
Chicago shot just 38.2% from the field, but had 19 offensive rebounds. The Lakers hit 42.2% of their shots with just nine offensive boards.
Kendall Marshall scored just three points with eight assists and four turnovers.
Playing on the second night of a back-to-back without five injured players (Kobe Bryant, Steve Nash, Steve Blake, Jordan Farmar and Xavier Henry), the Lakers fall to 16-26 on the season. Chicago, playing without Derrick Rose (knee), improved to 20-20.
The Lakers will travel to Miami to play the Heat (29-12) on Thursday. It will be their fifth game of a seven-game road trip.
The Bulls visit the Cleveland Cavaliers (15-26) on Wednesday.
Nick Young hit three free throws with 4.1 left on the clock to force overtime.
Joakim Noah had a chance to give the Bulls a four-point lead with 7.4 seconds left, but split a pair of free throws, opening up the door for Young's game-saving opportunity. Noah missed a difficult shot at the buzzer, leading to overtime.
Through regulation, Nick Young led all scorers with 26 points. D.J. Augustin had 22 for Chicago. Gasol contributed 18 points, 14 rebounds, five blocks and three steals.