The 20-year-old Tomic dominated the first set, but the big-serving Anderson won 24 of 26 points on his serve in the second and played more aggressively than his opponent as he tried to get back into the match.
Tomic had three unforced errors in the tiebreaker, including a floating backhand that went long on set point, sending the match into a third set.
Tomic broke Anderson at 4-3 in the tiebreak shortly after the South African had called for a fresh pair of shoes from the locker room, and then held serve to close out the match at Olympic Park.
"It's an amazing feeling," an emotional Tomic said after receiving the Ken Rosewall Trophy from the Australian legend.
"It's an honor to have won my first tournament in front of you," Tomic said, referring to Rosewall. "It's been a long road, but finally I have one of these trophies."
With Saturday's win Tomic's ranking will climb more than 20 places, possibly as high as No. 43, when the Australian Open begins on Monday, reclaiming the Australian top spot over Marinko Matosevic.
He is the ninth Australian winner of the Sydney event and first since Lleyton Hewitt in 2005.
The win was Tomic's eighth straight and completed a remarkable Open preparation. In addition to his win over Djokovic at the Hopman Cup, he later beat Tommy Haas and Andreas Seppi in Perth before victories against higher-ranked players Matosevic (47), Florian Mayer (26), Andreas Seppi and Anderson (36) in Sydney.