Returning to complete Tuesday’s unfinished rain-hit match, the Indian youngster conceded only two games, after the match resumed at 4-4, to win 7-5, 6-1.
He could have met compatriot Ramkumar Ramanathan in the next round, but the latter was beaten by Briton Brydan Klein 3-6, 3-6.
Yuki is now the sole Indian left in the singles event as Saketh Myneni lost 2-6, 1-6 to seventh seed Briton Liam Broady earlier.
In doubles, Myneni and Ramkumar are in the quarter-finals with their respective partners.
Myneni and Bulgaria's Dimitar Kutrovsky knocked out the top seeded Australian pair of Alex Bolt and Andrew Whittington, winning 7-5, 5-7, 10-7 in the first round.
Ramkumar and his Australian partner, John Millman, routed American wild cards Sam Barnett and Jesse Witten 6-0, 6-0.
Image: Yuki Bhambri
Photograph: Hannah Peters/Getty Images