The 27-year-old Murray, appearing on the newly refurbished Margaret Court Arena, did not look too troubled by the former junior world number one, even though he had some nervous moments in the third set when the qualifier held a 4-1 lead.
"The last couple days have been pretty cold, very cloudy, and today obviously the sun was out (and) the court was livelier," Murray said of the conditions at Melbourne Park.
"When it's dry as well, it didn't feel humid at all, the ball travels through the air quicker.
"He was hitting the ball very flat and coming to the net a lot (and) rushing me.
"I had never seen him play before, and never practiced with him. It was tough to know what to expect."
Bhambri is ranked 317th in the world and was making his grand slam main draw debut though Murray did not expect the 22-year-old to be languishing in the lower reaches of the rankings for too long.
"He's way better than ranked 300 in the world," Murray added. "He missed five or six months last year through injury.
"I would imagine by the end of this year he would definitely be around a 100 in the world if he stays injury-free.
"He returns well. He sees the game very well. He's talented. Good volleys. He likes to take the ball early and come forward.
"If he continues playing like that, I think by the end of this year he won't be far off."
The 2012 U.S. Open and 2013 Wimbledon champion, who is on the same side of the draw as Rafa Nadal and Roger Federer, will face Marinko Matosevic in the next round after the Australian beat Russian qualifier Alexander Kudryavtsev.
"I know Marinko very well," Murray said. "He's a good ball striker. Big guy. Serves well. Yeah, (it will) be a fun match.
"I don't always look at my draws.
"A lot can change in a couple of days in sport. You have just got to concentrate on your own matches, take care of business, and that's it."
(Editing by John O'Brien)
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