Kapur was the only Indian to fall inside the cut line as Jeev Milkha Singh (75-69) and Gaganjeet Bhullar (77-74) exited early from the tournament.
The 31-year-old Kapur, who has made the cut in each of his four starts on the 2013-14 European Tour season, looked well set at one point of time when he was four-under for the day after 13 and had avoided any bogeys till then.
McIlroy till then shared the top spot with American Brooks Koepka (65), whose flawless seven-under round put him at 10-under.
McIlroy, who went through his front nine in two-under 33, went to 12-under for the tournament with a birdie on 10th. Then came the two bogeys before he hauled himself back to the top on the 18th. Koepka at 10-under was sole second.
Tiger Woods (73) had a tough day with just two bridies against three bogeys and was way down in tied 44th place and made the cut with just one shot to spare.
"I played really well, even though it was windy when I played my round. At four-under with five holes to go, I felt I could do better, but I dropped that bogey on seventh where I went into the greenside bunker and then hit it to seven feet and missed the putt," Kapur said.
"A pity about that last bogey. But with morning conditions tomorrow I hope to go low and go up on the leaderboard," he added.
Jeev had five birdies, but will rue his two bogeys as he missed the cut by two. Bhullar, meanwhile, just could not get going and did not have a single birdie in his 74.
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