The duo was tied fourth, and two shots behind leaders, Sattaya Supupramai of Thailand and George Gandranata of Indonesia, who shot 66 each, while veteran Mardan Mamat carded 67 in third place.
Rashid, fresh from his top-10 finish and riding the confidence of having won in the earlier part of this year had four birdies on the back nine, which was his first nine, and had nine straight pars on the return journey.
Two other young Indians, S Chikkrangappa (69) and Chiragh Kumar (69) were tied for 11th place alongside Major winner, Jason Dufner.
Young Shubhankar (71), who was in contention last week at Panasonic Open India, and Abhijit Chadha (71) were tied 34th. Chadha closed with an eagle, while Shubhankar had three birdies against two bogeys.
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Jeev Milkha Singh (72) was tied 48th, while Rahil Gangjee, who lost in a play-off last week at Panasonic Open is tied 74th at two-over 74.
Rashid, who had two chip-ins from 30 and 40 feet on 13th and 16th, said, "It is nice every time you see your name on the leaderboard. My name was on the leaderboard early in the round and I told myself to keep it there. The course is playing very difficult this year. I remember shooting one-under last year and missed the cut."
"I'm very calm on the course now because I started working with a mental coach in India for a month. She (Aparna Jain) told me that I'm too aggressive on the golf course and that I need to be calm and play easy. She said I shouldn't put pressure on myself. It is a four-day tournament and I should take it step-by-step.
Leaders Sattaya and Gandranata both had seven birdies and one bogey each, while Mamat had seven birdies but two bogeys.