McSweeney took 17 balls to get off the mark and had a life on three when dropped by wicketkeeper Rishabh Pant.
But he made the most of his fortune, showing maturity in his second Test, while Labuschagne survived a nervy start as he targets a big score after a lean spell.
“It’s been my role for a while to attack the stumps and try and make early inroads,” said Starc. “Good way to start a Test. A good day all up.
“To finish the day just one down... to fight through that sustained pressure from a quality bowling attack is fantastic.”
India came into the clash riding high after a 295-run victory at the first Test in Perth, but Australia have a formidable record in Adelaide, winning all seven pink-ball Tests they have played at the venue.
The visitors retained Jaiswal and KL Rahul as openers in preference to Rohit, who missed the first Test on paternity leave and came in at six.
But the plan backfired as Starc sensationally trapped Jaiswal lbw from a swinging first ball, with the left-armer’s pumped-up celebration showing how much it meant.
Jaiswal scored 161 in his last innings.
The returning Shubman Gill then stroked two boundaries in an eventful first over.
But the runs dried up and Rahul was at the crease for 40 minutes without scoring when he was given out, caught behind off Scott Boland.
Rahul was walking off, but a no-ball was called and he survived.
Five balls later and he had another let-off, dropped by Khawaja.
Undeterred, he battled on before Starc wove his magic again to remove him for 37.
Virat Kohli roared back to form in Perth with an unbeaten century but he was no match for Starc this time, out for seven when Steve Smith collected a slip catch.
Gill weathered the storm, but his luck ran out just before the first break when he fell lbw on 31 to the relentless Boland.
‘Still in the game’
“I thought Australia did well, they set us up nicely with some of the dismissals, but also I don’t think 86-1 is a true reflection of how we bowled,” said India bowling coach Morne Morkel.
“I know the score looks like a big gap between the two teams, but we still feel we’re in the game.”
Six wickets tumbled in the second session, including Rohit (3) and Pant (21).
Pant should have gone for five, but McSweeney spilled a sharp chance in the gully off Boland, in the side for the injured Josh Hazlewood.
But the 35-year-old got his reward four balls later, beating Rohit’s bat to trap him lbw.
Pant fell to a vicious climbing ball from Pat Cummins before a hostile Starc returned and took care of Ravichandran Ashwin (22) and Harshit Rana (0) in the space of four balls.
With India in trouble, Nitish Kumar Reddy took charge, clobbering two big sixes in one Boland over for his highest Test score of 42 before he and Bumrah fell to Starc and Cummins respectively.
