The 31-year-old Jack received winning scores of 116-112 and 115-113, while the third judge scored it a 114-114 draw.
Jack, who fights out of Las Vegas, easily won the ninth round and appeared to be the fresher of the two fighters in the later rounds.
Jack improved to 19-1-1 while Dirrell dropped to 27-1-1.
"I was in great shape and I believed in myself," Jack told reporters after the fight. "All that talk don't matter. I knew they couldn't rob me. I thought I clearly won. It was a tough start to my career, but now I'm a world champ."
Jack has won three consecutive fights after being stopped in the opening round by Derek Edwards last year.
American Dirrell won the title in August with a hard fought decision over Sakio Bika.
Dirrell, 30, had to put his boxing career on hold for 20 months after being diagnosed with non-Hodgkins lymphoma.