She is one of two Americans who eliminated seeded players from Europe in the first round this time: Melanie Oudin, a surprise quarterfinalist at the 2009 U.S. Open, defeated No. 28 Tamira Paszek of Austria 6-4, 6-3 on Monday.
Five U.S. women were in second-round action Wednesday, and the two who are seeded won: No. 1 Serena Williams and No. 29 Varvara Lepchenko, who beat Elina Svitolina of Ukraine 7-6 (5), 6-1.
Unseeded Mallory Burdette lost track of the game score at one juncture and lost her match to No. 4 Agnieszka Radwanska 6-3, 6-2. Also exiting: Madison Keys and Shelby Rogers.
As her match against Safarova pushed deep into the final set, Hampton drew upon confidence she gained by winning three three-setters last week en route to the semifinals on red clay at a tournament in Brussels.
"I have those memories that I can pull from last week and they're very fresh and they're very recent," she said.
Safarova praised Hampton's resolve at key moments during their 2½-hour match, and also the American's touch, which led to three winners via drop shots.
"My game has a few more dimensions than other players', so that gives me a few more options, even when clay is not my best surface or it's not my favorite," said Hampton, who cut her left knee on a tumble Wednesday. "If I'm not hitting one shot great, then I can go to Plan B."
Up next is a second-round match against Anna Karolina Schmiedlova, an 18-year-old qualifier from Slovakia who is ranked 150th.
Hampton said she'd like to move her ranking into the top 30 by the end of this year, which would allow her to be seeded at Grand Slam tournaments.
"I'm looking to beat good players, beat top players, and move on in the Slams," she said. "Get deeper in the tournament, for sure."
In other words, make a name for herself.