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.
The difference showed in an opening set where Boulter struggled to land first serves and paid the price as she was broken three times, undoing the good work she had done to twice take Sasnovich’s unorthodox serve.
The Belarusian’s eye-catching motion, striking a pose like a flamenco dancer with her serving arm aloft before tossing the ball, looks awkward and lacked ferocity.
Boulter was able to break again at the first attempt in the second set which, in conjunction with more solidity in her service games, allowed the Briton to roll through the second set.
After saving a break point to hold for 5-2, Boulter took her opponent’s serve again to set up a deciding set.
Boulter had to come through a testing opening game, saving four break points to hold and dampen Sasnovich’s spirit before confidently racing through the final five games.
“I feel I came out really well but she responded quickly in the first set,” said Boulter.
“I tried to reset and play a better second and third and I’m proud I managed to stay in there and bring a good level to the court.”