Australia appeared poised to take a huge first-innings lead when India began day three on 5-164, still trailing by 310.
But the tourists lost just four wickets on Saturday as Reddy announced himself as a young star of world cricket by hitting his maiden Test century.
Reddy, just 21 and playing in his fourth Test, will resume on day four on 105, with India 9-358 and trailing Australia by 116.
Washington Sundar and Nitish Kumar Reddy hauled India back into the fourth Test on day three. (James Ross/AAP PHOTOS)
Australia made all the early running in this fourth Test, but India turned it around through Reddy and Washington Sundar (50) on Saturday to have their best day since dominating the series-opener in Perth.
Boland, the pick of Australia's bowlers with 3-57 from 27 overs, said team morale remained high.
"We're still leading by 116 runs, so pretty strong position," the 35-year-old said.
"Obviously, it could have been better, but I think that's the Test match ebbs and flows.Â
"Hopefully (Sunday) morning we get that final wicket, then put on a really nice lead and see how the game plays after that."
Mitchell Starc's back is causing concern as Australia strives to regain the Border-Gavaskar Trophy. (James Ross/AAP PHOTOS)
Australia have already bowled 116 overs as the effects of a gruelling series appear to take their toll on Mitchell Starc.
The star quick received treatment on his back on day three, but continued to bowl at top pace for the final session.
Starc has toiled for 25 overs in this Test, backing up from the 24 he bowled in Brisbane.
Boland was recalled to the XI after star quick Josh Hazlewood broke down for a second time in the series at the Gabba.
The fitness of Australia's bowling attack will be a major factor in deciding the outcome of the Border-Gavaskar trophy.
This Test is set to go five days, a rarity in modern cricket in Australia, meaning the players will have only two days off before the finale at the SCG gets underway.
With the five-match series tied 1-1, India will retain the Border-Gavaskar trophy if they pull off an extraordinary comeback victory in this Test.
Australia need to prevent the tourists from winning one of the last two matches of the series, while claiming at least one themselves, to regain the trophy for the first time since 2014-15.
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