World No1 claims 6-2 6-1 6-4 to stroll into the quarter-finals in New York.
Andy Murray plays his fourth round match against Grigor Dimitrov on 5 September.
Rafael Nadal has not reached the quarter-final of a Grand Slam event for the first time in 12 years.
The 13th seed lost 6-4 7-5 against the Latvian at Flushing Meadows.
Novak Djokovic plays his fourth round match against Kyle Edmund on 5 September.
Dan Evans came close to upsetting Stan Wawrinka in the US Open 2016.
Rafael Nadal plays his fourth round match against Lucas Pouille on 4 September.
Stan Wawrinka is in action in the third round against Dan Evans on 3 September.
Andy Murray faces Paolo Lorenzi in third round of the US Open 2016 on 3 September.
Great Britain's players have had a good time in the US Open 2016.
Great Britain will have three men in the last 32 for first time since 1968 after Evans beats Alexander Zverev.
Rafael Nadal plays his third round match against Andrey Kuznetsov on 3 September.
Novak Djokovic will be action in round 3 of the US Open 2016 against Mikhail Youzhny on 2 September.
Broady is currently at number 82 in the WTA rankings.
The Briton will meet either Giles Simon or Paolo Lorenzi in next round.
British no. 1 joins growing list of players to struggle in the heat at Flushing Meadows.
Wawrinka and Giannessi are meeting for the first time in an ATP tournament.
Andy Murray has won six out of the seven encounters against Marcel Granollers.
The 13th seed required medical attention towards the end of the second set.
The 21-year-old Brit won the match 7-5 6-4 6-4 in New York City.
Rafael Nadal plays his second round match against Andreas Seppi on 31 August.
Novak Djokovic looks to continue his form in round 2 against Jiri Vesely on 31 August.
Murray beat Lukas Rosol, while Broady defeated fellow countryman Laura Robson in the first round.
The Brit has now lost six times in the first round of the US Open.
Stan Wawrinka opens his US Open campaign against Fernando Verdasco in the first round on 31 August.
The British No1 barely broke a sweat as she secured a 6-3 6-3 win.
Andy Murray opens his US Open campaign against Lucas Rosol in the first round on 31 August.
The 21-year-old roared into the second round with a straight sets win.
Juan Martin del Potro won the silver medal in the men's singles event at Rio 2016 Olympics.
World number one Novak Djokovic is seeking his third US Open title.