The Korean-American, in his rookie year on the Tour, was at level-par till the 14th hole but surged to the top of the leaderboard with a four-under-par 68 for a total of nine-under 135.
Joshi, 20, also continued to shine in his maiden Asian Tour event as a professional, carding a three-under-par 69 to stay ahead of compatriot Rashid Khan and Bangladesh's Siddikur Rahman in the $300,000 event at the Delhi Golf Club.
"I finished with four straight birdies, definitely wasn't expecting it," Kim said after his grand finish.
"When I made the first two birdies on 15 and 16, I thought to myself 'I'm seven under and I was two back'. If I could make one or two birdies coming in, I would be right in there.
"I did feel some nerves in the last two holes but I controlled it well enough and stayed focus.
"I'll need to keep it in play tomorrow. Today, I kept in the fairways a lot. As long as you hit fairways, the greens are not hard to attack. Just have to play smart and try to get the putter rolling."
Joshi, India's top-ranked amateur last year, reached the turn in 36 but a brilliant chip-in for an eagle on the 14th hole made his day.
"It's great. I hit the ball good although my putting wasn't great," Joshi said. "Honestly, I felt normal. I didn't feel any pressure.
"I just wanted to go out there and do the best that I could. I'm looking forward to the next two days. It's a good position to be in. I'm not looking at the score board as it'll put pressure on me. I just want to go out to play."
Siddikur repeated his first-round 68 to be tied third with Rashid who blazed round the course in eight-under 64. Another Indian, Rahil Gangjee, fired two eagles in his round of 66 to be a shot behind in fifth.
The halfway cut was set at 146 with 76 players making it for the final two rounds.