There was a lot on the line for both women in the all-British clash on court 13, and it was Broady who proved the stronger at the end of a tight contest.
The 26-year-old from Stockport fought back from a set down to claim her second victory at a major – the other coming at Wimbledon two years ago.
Robson, 22, had hoped to claim a first Grand Slam win since the US Open in 2013, since when her career has been derailed by a serious wrist injury.
“I will get over it and probably think that overall this was a pretty good tournament but right now it sucks,” said Robson.
Broady served superbly under pressure, saving eight of nine break points and firing down 16 aces.
Asked about any possible ill-feeling on court, Broady said: “I think I was just getting annoyed because she was doing the ball tosses a lot of times.
“I am sure she is upset to lose. I don’t expect that she is going to smile at me at the net when she has just lost in three sets, so I’m sure she’ll be fine tomorrow.”
Broady now faces fourth seed Agnieszka Radwanska of Poland in round two.
Leon Smith, GB Davis Cup captain, told BBC Radio 5 live sports extra: “Naomi is probably more match tight and confident, Laura is still getting the belief back.
“But it’s good to have another Brit into the second round of a main draw, even if one has to go out. It’s a great thrill having Brits getting through, it gives everyone momentum and plenty to talk about.”