Boulter has leapt up the rankings after a strong 18 months where she has stayed fit and caused problems for opponents with her powerful game.
Now, she is aiming to break another ceiling by reaching the second week of a Grand Slam tournament for the first time.
But coming into the US Open, Boulter admitted she was not as prepared as she might have been for the hard courts because of her participation in the Paris Olympics which was played on clay.
That experience meant she played just three completed matches on hard courts before starting her campaign at Flushing Meadows.
In contrast, Sasnovich was match sharp having come through qualifying without dropping a set.
An edgy start saw the 28-year-old Briton lose serve three times in the opening set before settling down to take control.
“It’s not easy coming out here to play your first round against someone who has been top 30 – she knows how to play tennis,” Boulter, who reached the third round last year, told Sky Sports.
“I knew it would be a battle, she’s played three matches on these courts.”