Yuki, seeded ninth in the qualifiers, prevailed 6-4 6-2 in the second and final qualifying round to make it to the season's first Master series event.
Bhambri, who was junior world number one, had won Australian Open boys singles title in 2009 and has played only at this Grand Slam in the main draw since turning Pro.
Ramkumar Ramanathan won his second round of Australian Open qualifiers in straight sets.
Bhambri, seeded 15th, overcame Canada's Bradely Schuner 1-6 6-3 6-4 in the opening round and will next take on Spain's Carlos Taberner.
India were left to battle it out in the Asian zone yet again after Ramkumar Ramanathan's defeat in the must-win fourth rubber as Denis Shapovalov steered Canada back to the elite Davis Cup World Group
Ramkumar Ramanathan fetched the crucial win but Yuki Bhambri made the day memorable even in defeat as India and Canada shared the honours on day one of the Davis Cup World Group Play-off tie.
Ramkumar Ramanathan made it to the main draw as a lucky loser after Gael Monfils withdrew due to illness.
Yuki Bhambri and Saketh Myneni, injured during the Uzbekistan tie, return to Davis Cup squad as veteran Leander Paes is dropped.
It was in the eighth game of the second set that the 22-year-old served an ace to finish the match.
India defeated Uzbekistan 4-1 to enter the Davis Cup World Group Playoffs. If India win their next tie they will enter the elite World Group.
Rohan Bopanna and debutant N Sriram Balaji were clinical in routing their rivals as India sealed the Davis Cup Asia/Oceania Group I tie against Uzbekistan by taking an unassailable 3-0 lead on Saturday.
The Indian Davis Cup team's non-playing captain Mahesh Bhupathi dropped Leander Paes from the squad against Uzbekistan, opting for Rohan Bopanna instead.
India will take on Uzbekistan in the Davis Cup Group 1 Asia/Oceania second round for a berth in the World Group Playoffs.
Senior Pro Leander Paes' place in Indian Davis Cup team will be up for discussion when the AITA Selection Committee meets on Monday to pick the squad for the upcoming Asia/Oceania Zone tie against Uzbekistan.
Yuki Bhambri and Ramkumar Ramanathan won their respective matches to notch up straight sets victory to power India to a 2-0 lead against New Zealand on the opening day of the Asia/Oceania Group I Davis Cup tie
The 474-ranked player put up a spirited fight against a far superior Benoit Paire but went down 3-6, 4-6 in one hour and 25 minutes.
Having lost crucial six months last season due to tennis elbow injury, Yuki Bhambri made a good impression early in the season as he literally toyed with Ramkumar Ramanathan, who disappointed with his lacklustre show.
Saketh Myneni would need a herculean effort to make a winning start to the new season after being drawn against Mikhail Youzhny in Chennai Open Round 1