France is captained by world number 55 Gilles Simon, and both Benoit Paire and Gael Monfils credited the 35-year-old with helping them overcome their opponents.
Roger Federer admitted he struggled to get past the talented Gael Monfils.
Ramkumar Ramanathan made it to the main draw as a lucky loser after Gael Monfils withdrew due to illness.
Bhambri, who had qualified for the main draw of the ATP 500 tournament, took one hour and 51 minutes to register the 6-3 4-6 7-5 upset win over the sixth-seeded Frenchman to enter the round of 16.
Novak Djokovic clinched his first title since January with a 6-3, 6-4 victory over Gael Monfils.
Gael Monfils had lost both his group stage matches at London's O2 Arena and, with no chance of making the semi-finals, the Frenchman decided not to risk aggravating his rib injury in his final match
Third seed Stan Wawrinka beat World No.1 Novak Djokovic to clinch his third Grand Slam title. Catch US Open tennis highlights here
The semi-final was clouded by controversy when Gael Monfils appeared to stop competing in the first set, standing inside the baseline, airily slicing first serves back to Novak Djokovic.
Although Novak Djokovic holds a 19-4 record over Stan Wawrinka, including two victories since the Swiss shocked him in the 2015 French Open final, Wawrinka welcomed the chance to take him on again.
Defending champion Novak Djokovic defeated Gael Monfils to reach the seventh US Open final of his career.
World No.1 Novak Djokovic takes a 12-0 career lead over Frenchman Gael Monfils into their US Open semi-final on Friday while Kei Nishikori will look to build on his shock five-set win over Andy Murray against Stan Wawrinka
Novak Djokovic will face another Frenchman, Gael Monfils, for a place in Sunday's US Open final after the 10th seed downed compatriot Lucas Pouille 6-4, 6-3, 6-3
Gael Monfils and Angelique Kerber both reached the US Open semifinals on Tuesday after securing easy wins in their singles quarterfinal matches
After the 6-3, 6-2, 6-3 defeat to Gael Monfils in the US Open fourth round, Marcos Baghdatis said he was actually composing a message to his wife, former WTA top-20 player Karolina Sprem
Rafael Nadal, who hasn't reached a major quarter-final since Roland Garros last year, was defeated in five sets by Frenchman Lucas Pouille in the US Open fourth round match
France's Gael Monfils has pulled out of the 2016 French Open due to a viral infection. Earlier, Roger Federer also announced that he would miss the tournament due to injury.
Rafael Nadal defeated Gael Monfils in a tough three-setter to win the Monte Carlo Masters for a record ninth time.
Rafael Nadal defeated Andy Murray 2-6, 6-4, 6-2 in Monte Carlo Masters to enter the final. Nadal hasn't won a Masters title since 2014.
