Kapur, who was four-over after the first five holes with two bogeys and a double bogey, rebounded strongly with seven holes over a nine-hole stretch to keep himself in line for a strong finish at the event sanctioned by the Asian Tour and PGA Tour.
Kapur is tied 12th, while India's other two challengers, Gaganjeet Bhullar (72) and Anirban Lahiri (74) are lying 30th and 54th, respectively in the 78-man field that has no cut.
India's other representatives in the CIMB Classic, which has attracted 18 of the world's top-50 players in the field including the top-10 players from the Asian Tour, endured mixed fortunes.
Bhullar, currently third on the Asian Tour's Order of Merit, battled to a 72 after finishing with two birdies over his last three holes while Lahiri, who was runner-up in the Venetian Macau Open last week, settled for a 74 after shooting two birdies, two bogeys and one double bogey.
More From This Section
He dropped two bogeys and one double in his opening five holes at the Kuala Lumpur Golf and Country Club's West course, before coming back.
"Obviously four over after five holes, I said to my caddie, 'look, I haven't really hit too many bad shots'. I had a bad lie in the rough on 14 and I should have probably played away from the water and made double there.
"And after that, I was swinging it well, hitting good shots. I just tried to get back to as close to par and once I did that, I just kept going and going and made seven birdies in my last 13 holes or in fact seven birdies in a stretch of nine holes," he said.