On Monday, Federer overcame a scare from Indian qualifier Nagal to win 4-6, 6-1, 6-2, 6-4 in a match that lasted two hours and 30 minutes.
It was for the first time since 2003 that Federer had lost a set in his first-round match at the US Open.
Before Nagal, Rohan Bopanna (7-6, 6-2 in 2006) and Somdev Devvarman (6-3, 6-3 in 2011 & 6-2, 6-1, 6-1 in 2013) were the other Ind to take on the five-time US Open champion.
"He showed incredible composure against the greatest player in the world and held his own," Bhupathi told .
"I think both he and his coaching team should be very proud of the improvements they have made this year," the Indian tennis great and Davis Cup non-playing captain added.
Meanwhile, Federer, who was sporting a light stubble, said in his on-court interview: "I thought I played like my beard today, I was rusty."
World No. 190, Nagal, walked onto Arthur Ashe Stadium for his first Grand Slam main draw match without having ever earned a tour-level victory.
Nagal sent heavily-spinning shots over the net and used the court well and won the first set 6-4 as Federer made 19 unforced errors.
However, in the next three sets, Nagal, seeking his first tour-level win, did not seem to have the weaponry to take the racquet out of the Swiss' hands and lost 1-6, 2-6, 4-6 to bow out of the competition.
Once Federer cut down his errors after the first set, he kept on top of Nagal, and recovered to enter the next round where he will face Bosnia's Damir Dzumhur.
Before the match, Nagal had said he would try to enjoy the game and not think too much about whom he was playing against.
"I want to play him. It doesn't matter. It's not about winning or losing all the time. It's just the experience. It's playing someone who has 20 Grand Slams," Nagal had told ATP Tour website.
"I don't care what the commentators will be saying about me on TV. I'm going to be enjoying the crowd, playing the best tennis player ever. I'm just some dude from India. I'm fine with that until I make my name. That's it," he added.
(Debayan Mukherjee can be contacted at debayan.m@.in)
( With inputs from IANS )