52 ranking spots separated the two but that hardly bothered the 24-year-old American. The reigning French Open champion failed to get any break point and had her serve broken only once which was enough for Brady to wrap up the opening set.
The second set was more even and it went to a tie-break where the American served strongly to bag the 6-4, 7-6(4) win in 1 hour 32 minutes.
Brady will next take on the fifth seed Petra Kvitova for a place in the semifinals.
The two-time Wimbledon champion was in sublime form and needed only one hour to breeze past 129th ranked qualifier Liudmila Samsonova 6-3, 6-2.
Both of them produced seven aces in a strong serving display but it was the Czech player's ability to convert all three of her break points was what turned the match in her favour.
While the fifth seed had an easy time on the court, it wasn't so for the rest of the seeds who won on Day 4 of this WTA Premier level tournament. Second seed Karolina Pliskova, third seed Naomi Osaka and sixth seed Kiki Bertens all had to work hard to complete their wins in three sets.
Bertens was involved in the toughest of the battles with World No.26 Anett Kontaveit in a blockbuster second-round encounter. The World No.9 Dutchwoman made a good start to the match, banking on her feisty groundstrokes and took the first set 6-3. The Estonian, however, came storming back and bagged the second set 6-2 to level the match.
What ensued in the decider was tennis of an absolute sterling quality. Both women refused to break down and it was finally the solitary break of serve that Bertens managed to earn, which ultimately got her the win in 121 minutes.
Bertens next faces two-time Grand Slam champion Osaka, who clawed her way back to a win after a slow start against rising star Sofia Kenin. The 21-year-old Russian sneaked the first set in a tie-break before the Japanese rebounded to take the last two sets for a 6-7(3), 6-3, 6-1 win.
Former US Open runner-up Pliskova too had to put in a lot of effort in her 121-minute marathon victory over Ajla Tomljanovic. The first set was a breakfest in which the Czech ace queen's two breaks of serve were enough to seal the set.
While the highly competitive second set went to a tie-break and was eventually won by the Australian, Pliskova made sure that her smooth, effortless groundstrokes had the last word in a remarkable show in the decider. The second seed ended the match in a 6-4, 6-7(5), 6-1 win to set up a quarter-final showdown with Alison Riske of the USA.