The Red beat Italian side Atalanta in the second leg quarter-final but lost 3-1 on aggregate as Leverkusen advance to semis despite 1-1 draw with West Ham
Indian openers captain Rohit Sharma and KL Rahul made a steady start as the hosts finished the first day of the second Test on 21 for no loss, after bowling Australia out for 263 on Friday.
Rohit was on 13, while Rahul was on four, at stumps.
Earlier, fast bowler Mohammed Shami took four wickets as India bowled out Australia for 263, with Peter Handscomb hitting 72 not out.
Handscomb and Usman Khawaja (81) both played gritty knocks to boost their side's total, after the tourists elected to bat following a drubbing in the opening match of the series.
Spinners Ravichandran Ashwin and Ravindra Jadeja each got three wickets on a turning wicket.
Shami wrapped up the Australian innings in the final session as he bowled left-arm spinner and debutant Matthew Kuhnemann for six. Handscomb scored nine boundaries in his 142-ball knock.
The left-handed Khawaja's innings came to an end before tea, with a stunning one-handed catch from KL Rahul giving Jadeja his 250th Test wicket.
Handscomb, who reached his fifth Test half-century, kept up the grind in a 59-run eighth-wicket stand with skipper Pat Cummins, who made 33 before being trapped lbw by Jadeja.
Handscomb had arrived at the crease with Khawaja in the second session, after Travis Head departed, the pair putting on 59 runs before Jadeja broke through.
Rahul's acrobatics at one point saw him dive full stretch to his right to grab the ball after a reverse sweep from Khawaja, who fell to his knees in disbelief.
Ashwin soon got wicketkeeper Alex Carey for a duck for his third wicket of the day as India once again called the shots after an Australian fightback.
The off-spinner had struck twice in one over before lunch, getting Marnus Labuschagne for 18 and Steve Smith for a duck to put the tourists in trouble before lunch.
David Warner, who scored one and 10 in Nagpur in the first Test, was the first to go after he got his first runs off the 21st ball he faced. Shami took his wicket.
He had faced a barrage of short balls by seamer Mohammed Siraj and took a couple of blows, on the elbow and the back of the helmet.
Just when it looked like the left-handed opener had settled in, Shami produced a peach of a delivery that pitched and held its line from around the wicket, with the keeper taking the edge behind.
Khawaja rode his luck with the ball taking a few inside edges off the bat but he battled through to raise his 20th Test half-century.
Test specialist Cheteshwar Pujara was presented with a special cap on his landmark 100th match and entered the field through a corridor created by the players.
The hosts have not lost a Test at the venue — previously called Feroz Shah Kotla — since 1987.
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