Barty and Dellacqua broke the Italian pair in the eighth game of the second set, then held serve in the next game to force a deciding third set. The Australian pair broke Errani and Vinci to start the deciding set, but the Italians recovered quickly and dominated the remainder of the match.
"In the third set we started to get nervous," Vinci said. "However, today we've proven we're a really strong doubles team.
"Our strength is that we always play together. We went out there today with lots of grit, and we really wanted to win."
The 16-year-old Barty was attempting to become the youngest Grand Slam champion since Martina Hingis won the Australian Open singles title in 1997.
"It felt like we were really close in that match, a lot closer than the score suggested," Barty said. "A point here and there, and I think that third set could have really turned.
"Obviously we're disappointed not to have had the win, but for me this has been just an unbelievable two weeks, an unbelievable tournament."