Starc triggered India's downfall in last week's first Test in Pune by claiming the wickets of Cheteshwar Pujara and Kohli in quick succession in the first innings before left-arm spinner Steve O'Keefe took charge.
Australia bowled out India for paltry totals of 105 and 107 in the two innings, with O'Keefe taking a total of 12 wickets as the visitors wrapped up a 333-run win inside three days.
Kohli, who has been in red-hot form in recent months, had a rare failure with scores of 0 and 13 in his two outings.
Guilty of chasing Starc's wide delivery outside the off-stump, Kohli gave away a catch to Peter Handscomb at first slip and the batting soon fell apart on a sharply turning wicket.
"We know (Kohli) is going to be another key wicket for the rest of the series. There's six more times we've got to get him out to really cement this series," Starc said.
"We know he'll come back bigger and stronger, but he's one I'll take over Pujara at the moment," Starc said when asked to rate the importance of his twin strikes in Pune.
Starc and Kohli are former team-mates for the Indian Premier League (IPL) side Royal Challengers Bangalore, and the Australian is well aware of how dangerous the Indian skipper can be once he gets his eye in.
"He's a class player, we all know that. He's scored a mountain of runs already this year. We're going to have to be wary of that Virat comeback," Starc said of Kohli, who has scored double centuries in four successive series.
"We are really happy that it happened that way for us, but we know that one Test is not going to win us the series. It's three important Test matches to go."
After the next Test in Bangalore, the two teams are also due to play in Ranchi and Dharamsala.
First Published: February 28, 2017, 5:17 PM IST