- Thakur earlier got the ball rolling for India with the first wicket of the day, having opener Marcus Harris caught behind for 38.
Australia stretched their lead to 276 runs in the fourth test decider against India in Brisbane on Monday before rain brought tea early with the hosts having reached 243 for seven.
Tailender Pat Cummins was two not out, with Mitchell Starc on one run at the Gabba, in a contest shaping up to be a cliffhanger if conditions allow.
With the series level at 1-1, the hosts must win the match to claim the Border-Gavaskar trophy while India need only a draw to retain the silverware after their breakthrough 2-1 win in the 2018/19 series in Australia.
The home side face a tricky decision as to when to declare with rain also forecast on the final day.
Australia resumed on 149 for four after lunch, with Steve Smith and Cameron Green looking to build the lead after India tore through the top order with 4-34 in the morning.
A Gabba pitch that was largely benign through the opening days began to bare its teeth, with Smith (55) gloving a catch to captain Ajinkya Rahane after a Mohammed Siraj ball reared up from the wicket.
India's bowlers kept chipping away and Thakur was rewarded with the wicket of Green (37), who edged to Rohit Sharma, the slips fielder's fifth catch for the match.
Thakur took his third wicket of the innings by banging in a short ball that captain Tim Paine (27) tried to hook but succeeded only in feathering an edge to a leaping Rishabh Pant behind the wicket.
Thakur earlier got the ball rolling for India with the first wicket of the day, having opener Marcus Harris caught behind for 38.
Debutant Washington Sundar trapped David Warner for 48, with Siraj having Marnus Labuschagne caught in the slips for 25.
Three balls later, Siraj had Matthew Wade caught behind for a duck.