Star batsman Kohli will begin Saturday's fourth day on 43 not out in a total of 110 for five, with India needing a further 84 runs to reach their victory target of 194.
"We'll go to bed dreaming about getting Kohli out first thing," said Anderson.
Kohli's magnificent 149 earlier in this match, his maiden Test century in England, meant India were only 13 runs behind on first innings when it looked as if they might be at least a hundred runs in arrears.
His century was also notable for the way he shepherded the tail.
Nevertheless, a bullish Anderson told reporters after Friday's close: "No one is invincible in world cricket. We can get him (Kohli) out.
"If we recover well tonight and come back fresh in the morning, we know it's going to be 25-30 overs max one way or the other –- so we can give it all we can."
But Anderson, England's all-time leading Test wicket-taker, admitted: "If he (Kohli) bats like he did first innings … even batting with the tail, it was so hard to keep them on strike, have six balls at them.
"He just managed to find the gap, or get down the other end –- the way he batted with the tail was brilliant.
"We need five wickets, simple as that, fairly quickly –- otherwise, they'll get the runs."
The Lancashire swing bowler added: "It's a really exciting place to be, because you know we could do something special tomorrow by winning this."
But the match position in the first of this five-Test series might have been so different had not Kohli been dropped in the slips by Dawid Malan on 21 and 51 en route to his hundred in the first inningh.
"It's an area where we've struggled for two years now," said Anderson, the unlucky bowler when Malan missed Kohli the first time, of England's poor close-catching.
"You can't drop Virat Kohli on 21, because he's one of the best players in the world, and he'll capitalise on that –- which he did. All we can do is work extremely hard at it, and hope to improve."
Meanwhile India fast bowler Ishant Sharma, who took an impressive five for 31 in England's second innings 180, backed Kohli and Dinesh Karthik (18 not out) to see his team to victory.
"It gives us a lot of confidence but at the same time Karthik's positive intent has also given us confidence," said Sharma.
"Losing five wickets doesn't mean much...Cricket is such a game that you can chase down small totals easily with one big partnership."
England's 1,000th Test featured an all-too familiar batting collapse as they slumped to 87 for seven on Friday, with Sharma taking three wickets in an over.
But they avoided a humiliating loss thanks to No 8 Sam Curran's rapid 63, his maiden Test fifty.
That innings followed the 20-year-old Surrey's left-arm swing bowler's Test-best four for 74 on Thursday in what is just his second match at this level.
"I remember me as a 20-year-old, and I didn't know what was going on," said Anderson, now aged 36.
"He (Curran) knows exactly what he's doing, how to set batsmen up, and we saw he has talent with the bat as well.
"To go out in that situation and show the guts and determination he did to get the team back into the match, when it looked like we were down and out, I thought was brilliant from such a young guy," he added.
Curran said he'd followed the advice of England captain Joe Root.
"It was a tricky situation but Rooty had a word and said just play my natural game," Curran told BBC Radio's Test Match Special. "I didn't want to overthink it."
First Published: August 4, 2018, 8:21 AM IST