With India losing both their openers cheaply on a baking hot morning at Adelaide Oval, Kohli drove hard at a Pat Cummins delivery to be caught in the gully for three by Khawaja, who leaped at full stretch to pluck the ball with his left hand.
Having won the toss and elected to bat on a day when the temperature was forecast to reach 39 degrees Celsius, India were reduced to 19 for three before the drinks break.
Seamer Josh Hazlewood, who bowled superbly in his first spell, turned the screws further with his second wicket of the morning, removing Ajinkya Rahane for 13.
Rohit Sharma, recalled at the expense of all-rounder Hanuma Vihari, survived a nervous period before lunch to reach the break on 15 not out, with number three Cheteshwar Pujara on a watchful 11 from 62 balls.
While plenty of pessimism has surrounded Tim Paine's team, Australia's pacemen were quick to remind India of their quality on home pitches.
Hazlewood captured the first wicket of the series with his sixth ball, coaxing the out-of-form Lokesh Rahul into a loose drive away from his body. The nick flew to Aaron Finch at third slip and Rahul trudged off for two.
Pace spearhead Mitchell Starc then removed Murali Vijay for a streaky 11, the recalled batsman edging another rash shot behind to wicketkeeper-captain Paine.
India's usual saviour Kohli strode to the crease with an average above 98 in Adelaide but survived only 16 balls before falling to seamer Cummins's third delivery of the day.
Khawaja, the man who caught him, had taken his place in the side despite the arrest of his brother on charges of attempting to pervert the course of justice and forgery on Tuesday.
Australia went in to the test with six specialist batsmen, having dropped vice captain and all-rounder Mitchell Marsh in favour of Peter Handscomb.
Former Australia batsman Mike Hussey presented debutant opener Marcus Harris with his baggy green cap before play.
Kohli is hoping to lead India to their first series win in Australia, having played a starring role in their last trip Down Under when Mahendra Singh Dhoni's team lost 2-0 in 2014/15.
The series continues after Adelaide with tests in Perth, Melbourne and Sydney. (Editing by Nick Mulvenney)
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