Collins came out all guns blazing in the opening set to go up 3-0 and held firm to take the early advantage, but Barty's battling qualities surfaced in the next as she stretched the rallies to claw her way back into the contest.
The pair then exchanged a couple of breaks in the deciding set before the tie-break and Barty appeared to feed off the home fans' energy to secure victory on her second match point.
"It was tough but it was a lot of fun," Barty said during a courtside interview, before addressing the crowd.
ATP Rome doubles
Murray and Broady bow out of ATP Rome Doubles
"You guys have brought a little bit more enjoyment back into my tennis ... I can't wait to get back out here tomorrow."
Top seed Barty, who lost in the round of 16 in Brisbane, said she was slowly finding her best form ahead of the Australian Open which begins on Monday.
"It's new beginnings ... Brisbane was tough, I didn't feel I was quite sharp enough to win matches there ... but I'm getting better and better," she added.
Up next for the French Open champion is teenager Yastremska, who beat sixth seed Aryna Sabalenka 6-4 7-6(4) to move into the biggest final of her career.
"It's nice. I'm very happy," said world number 24 Yastremska, who is looking to capture her fourth WTA title. "Especially in the beginning of the year. It brings me some confidence before the Slam."
Barty won her only previous meeting with Yastremska in Miami last year, and will look to head into the season's first major with an eighth singles title in the bag.
She faces Ukraine's Lesia Tsurenko in the first round of the Australian Open where she will bid to end the host nation's 42-year wait for a home-grown champion.
In the men's tournament at Adelaide, South Africa's Lloyd Harris reached his maiden ATP Tour final beating American Tommy Paul 6-4 6-7(3) 6-3 on Friday.
The world number 91 will next take on Qatar Open champion Andrey Rublev who edged past Canada's Felix Auger-Aliassime 7-6(5) 6-7(7) 6-4 in the other semi-final.
Barty powers past Kudermetova to reach Rome quarter-finals
Konta suffers swift defeat to Ostapenko at Italian Open