Third seed Bhambri pipped Spain's Pedro Martinez 6-2 7-6 (0) while Nagal battled past Great Britain's Brydan Klein 6-4 4-6 7-5 in a marathon quarterfinal clash.
In another match, Prajnesh defeated Australian Marc Polmans 6-2 6-7 (1) 6-1.
However, last week's runner-up and fifth Ramkumar was ousted following a 7-6(3) 2-6 4-6 defeat at the hands of Briton Jay Clarke.
Before taking on Polmans, Prajnesh had stunned British sixth seed Evan King 6-4, 6-4 in straight sets, avenging his loss to him at KPIT-MSLTA Challenger in Pune.
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After winning the first set, Nagal lost his tempo and conceded the second set to the British, who powered his way through with aggressive tennis.
Klein stretched the fight in the third and final set, but crucial points to suffer defeat.
Meanwhile, Taipei's Tsung Hua Yang trounced Croatian Borna Gojo 2-6 6-4, 6-2. Slovenian Kavcic defeated Mario Vilella Martinez of Spain 6-4, 3-6, 6-3.
Ante Pavic caused the biggest ripple as he showed fourth seed Elias Ymer the door. The Croatian, who won 6-3, 6-7(2), 7-6(5), will clash with Clarke in the quarterfinal.
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