Italian pair Errani and Vinci have now won three of the past four major titles, with the Wimbledon crown the only one they do not currently hold.
Australians Barty and Dellacqua, who entered the tournament as a wildcard pairing, enjoyed the backing of a home crowd but ultimately fell short in the final set.
The result means that Barty fell short in her bid to become the second-youngest Grand Slam champion in history at just 16 years of age.
Top seeds Errani and Vinci raced into the lead early on Rod Laver Arena as Barty and Dellacqua dropped their first set of the tournament.
The Aussies fought back to take the second set and level the contest but could not keep up with their opponents in the decider as Errani and Vinci captured a break at 3-2 and did not look back.
"In the third set we started to get nervous. However, today we've proven we're a really strong doubles team," Vinci said.
"Our strength is that we always play together. We went out there today with lots of grit, and we really wanted to win."