Kwame Brown added 10 points and 10 rebounds for the Lakers, who have won a season-best seven in a row.
Seattle led 121-117 after Kurt Thomas hit a shot with 3:03 to play. But after Dallas missed two free throws, Bryant hit a short jumper and then followed missed Dallas shots with a bank shot and another jumper to put the win in the Lakers' column.
Durant nailed a three-pointer with 46.3 seconds left to force overtime.
Notable deaths of 2014 [PHOTOS]
Arizona Cardinals player formally charged