The match, which started just after midday on one of the balmiest days in Key Biscayne, proved to be as much of a test of fitness as it was a test of Brengle and Radwanska's tennis.
"Today was really hot. And yes, we both kind of have to get used to [the humidity]," Radwanska said after the match. "Today was a really tough day, and I knew that since this morning.
"You just try everything to stay cool, especially during the changeovers."
It was Brengle, the unseeded American making her Miami third round debut, who kept her cool first, breaking Radwanska's serve early and consolidating for a 3-1 lead.
But her efforts weren't enough to trouble the World No.2, who quickly got the break back and leveled the score. A handful of well-crafted points finished at the net gave Radwanska the advantage, and she grabbed the next three games to take the set.
? WTA (@WTA) March 26, 2016
The pair hung tight to start off the second set, trading holds and breaks of serve for 2-2. But the heavy conditions began to take their toll on the American, whose shots misfired on the important points allowing Radwanska to rattle off four straight games for a comfortable win, 6-4, 6-2.
Radwanska is into the Miami fourth round for the eighth time in her career, and she's seeking to extend her impressive run of reaching the semifinals or better in eight of her last nine events, a streak dating back to October of last year.
Though after an hour and 13 minutes in this intense heat, Radwanska is looking forward to a more immediate reward:
"One second after the match point, all I'm thinking is: the ice bath!"
In a see-saw opening, Ivanovic recovered from losing four games on the trot to hold a set point at 5-4. However, a wild forehand let Bacsinszky off the hook as the momentum swung again. The following game, the former World No.1 wavered, double faulting to surrender her serve and ultimately the set.
After struggling at the start of the year, Bacsinszky has been quietly playing herself into form in recent weeks, reaching the last 16 in Doha and Indian Wells, and despite a late rally from Ivanovic she held on the extend this streak.
Also advancing in this quarter were Simona Halep and Heather Watson. No.5 seed Halep withstood some early resistance before easing past Julia Goerges, 6-4, 6-1, while Watson dug deep to overcome Yanina Wickmayer, 3-6, 7-5, 6-3, in a match lasting over two and a half hours.