Chasing a victory target of 143, India were down two wickets for 11 runs in the third over, and the required run rate quickly shot past nine per over.
Raj and captain Jhulan Goswami dragged the visitors back into contention in the final eight overs with an unbroken 60-run partnership at Hurstville Oval.
The experienced Indian duo steadily built their innings and needed a six off the final ball to tie the match and force a shootout, but Raj could only manage a boundary to give the home team a hard-fought win.
Raj's 51 came of 45 balls. Apart from Raj the other top scorers for India include Sulakshana Naik (32) and Goswami (27 not out) as India mounted a spirited run chase only to end just short of the target.
Earlier, a dazzling innings of 44 from 23 balls by captain Karen Rolton guided Australia to 142 for four after being sent into bat.
Despite suffering a thigh injury early in her innings, Nitschke (49 from 40) was the other main contributor with the bat for the hosts. She batted with the aid of a runner and hit four boundaries and two sixes.
Rolton, later adjudged player-of-the-match, smashed eight boundaries on way to her 44.
Australia and India start their five-match One-Day International series on Friday at Hurstville Oval.