Bhambri, who had qualified for the main draw of the ATP 500 tournament, took one hour and 51 minutes for a 6-3 4-6 7-5 upset win over the sixth-seeded Frenchman.
The Delhi-based Bhambri had beaten world number 16 Fabio Fognini in 2014 Chenni Open but the Italian had retired from that match due to a fitness issue.
"It's incredible," Bhambri told PTI.
"I just tried to fight for every point and focussed on holding my serve. I played aggressive and took my chances which
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It is also only the second time that Bhambri has won two main draw matches at an ATP World Tour event since his quarter-final appearance at Chennai Open in January 2014.
The last time Yuki toppled a top-50 player was in September 2015 when he beat Czech Republic's Jiri Vesley.
He has enjoyed wins over top-100 players -- Japan's Go Soeda (ranked 87 in 2015) and Jack Sock (ranked 88 in 2013).
Asked what it takes to beat top players, Yuki said, "There is no formula."
"You have to treat it like another match and you have to believe and trust your abilities. I have always said I have the game, for me it's about staying healthy and this win gives me confirmation to the fact that I can compete with the best," Yuki said.
Pella, who had shown the door to another Indian Ramkumar Ramanathan in the opening round, defeated Mischa Zverev of Germany 6-7(5) 7-6(2) 6-3 in the second round.
It was good news for India in the men's doubles as well with Rohan Bopanna and his partner Donald Young of USA overcoming their opening hurdle.
Wildcard entrants Bopanna and Young shrugged off the challenge of Canadian Daniel Nestor and Aisam-ul-Haq Qureshi of Pakistan in straight sets.
Bopanna and Young hardly broke a sweat to get the better of Nestor and Qureshi 6-2 6-3 to seal their place in the quarterfinals.