Sutherland (3-38 off 14 overs) and 19-year-old debutant Lucy Hamilton (3-31 off 11) were the chief destroyers in the 36C heat on Friday.
But India's innings would have been wrapped up before dinner if not for a series of dropped catches in the field from Australia.Â
Hamilton snared the first wicket of the day when she bowled Smriti Mandhana for four in the sixth over, sparking wild celebrations.
Wickets fell at regular intervals after that, but Australia's butter fingers proved costly.
Sutherland had four catches dropped off her bowling.
The first came when Hamilton grassed Indian debutant Pratika Rawal at gully.
Rawal, who was on four at the time, went on to make 18.
Captain Harmanpreet Kaur, who was dropped at gully on 10, went on to make 19, but the most costly missed chance was when Jemimah Rodrigues played a shot to Georgia Voll at short leg.
Voll couldn't hold the one-handed chance when Rodrigues was yet to score.
The India batter went on to make 52 off 84 balls, with her knock including four consecutive boundaries from leg-spinner Alana King's first over of the day in the 40th over.
Hamilton eventually had Rodrigues caught at square leg, but her half-century ensured there was no dire crumble from India.
Sutherland watched a fourth catch off her bowling go down when wicketkeeper Beth Mooney had three juggles before dropping an easy edge from Kashvee Gautam.
Gautam was on 16 at the time, and she made it to the dinner break unbeaten on 22, alongside fellow debutant Sayali Satghare (7no).
Star all-rounder Ellyse Perry, who missed all three ODI matches of the multi-format series with a quad niggle, was selected for the Test, but as a batter only.
Hamilton, 19, was included after Kim Garth was ruled out with a quad injury, while incoming captain Sophie Molineux (back) was another big name to be ruled out.
The four-day match is the final international appearance for Australia skipper Alyssa Healy, with the 35-year-old slated to bat at No.4.
Australia lead the multi-format series eight points to four, meaning India must win the Test to come away with a draw.
