With the hosts needing three to reach their target of 275, India thought they had won when Nicola Carey was caught.
But the third umpire ruled Goswami’s full toss to be above waist height.
With two needed from the resulting free hit, Carey gave Australia a 2-0 series lead and their 26th straight win.
That winning streaks is a record in one-day internationals for both women and men.
The late drama came at the conclusion of a superb contest in Mackay, Queensland.
India made 274-7, their highest total against Australia in Australia, thanks mainly to Smriti Mandhana’s 86.
Australia were reduced to 52-4, only for opener Beth Mooney to rebuild in a stand of 126 with Tahlia McGrath, who added 74 to the 3-45 she claimed with the ball.
Carey joined Mooney, who ended on 125 not out, for an unbroken partnership of 97, with 13 needed from the final over.
Goswami had already hit Carey on the head with a beamer from the third delivery, but India thought the final delivery was legal.
After an agonising wait, it was adjudged to be a no-ball, leaving Carey to shovel the final delivery to long-on for the winning runs and end unbeaten on 39.
India 274-7 in 50 overs (Mandhana 86, McGrath 3/45)
Australia 275/5 in 50 overs (Mooney 125*, McGrath 74) (BBC)