Green enabled Australians' recovery from 60/3 to 286/8 by the close of play.
After the Indians declared at 247/9, the home team found themselves in trouble at 60/3 when Green walked up to the crease. The score soon became 98/5.
Read Also: "Haven't faced quality like that", Cameron Green lauds Indian spinners
The Western Australia all-rounder, however, batted beautifully in a gutsy 104-run stand with Tim Paine (44) and soon brought up his century to also strengthen his claim of a Test debut against India.
Green got another able ally for support after Paine's departure in Michael Neser (33) who held his end safely and made some really crucial runs to put Australia ahead along with his more capable partner.
Earlier on, Indian quicks Umesh Yadav (3/44) and Mohammed Siraj (2/71) built inroads with the new ball. But they'll be disappointed not to have carried on and run through the batting order, allowing Green & co to pull off their recovery act.
Nevertheless, the two bowlers still had a nice workout ahead of the Test matches in what is effectively an audition to cement the third seamer's slot in Ishant Sharma's absence.
Off-spinner Ravichandran Ashwin (2/58), who looks set to play the first Test with uncertainty around Ravindra Jadeja's hamstring niggle, also bowled close to 20 overs and got among the wickets.
Brief Scores:
Australia A 286/8 in 85 overs (Cameron Green 114*, Tim Paine 44; Umesh Yadav 3/44, Ravichandran Ashwin 2/58) lead Indians 247/9 decl. (Ajinkya Rahane 117*, Cheteshwar Pujara 54; James Pattinson 3/58, Travis Head 2/24) by 39 runs