Stephens was in a dangerous situation midway through the match - having dropped the first set, 6-4, she found herself down break point serving at 5-5 in the second set - but she saved that and took the second set, 7-5, and after losing serve in the opening game of the third the No.17-seeded American was just too solid for a somewhat erratic Puig, running away with an emphatic 46 75 61 victory.
"Monica was playing really well," Stephens said. "She was going for all of her shots. It's tough when you're playing someone who is just going for everything. It's kind of hard to get into a rhythm. But I just hung in there and tried to do my best. Once I got my opportunity in the third set, I just went for it."
It hasn't been a smooth ride to the quarterfinals for Stephens by any means - she was pushed to 8-6 in the third set by Andrea Petkovic in the second round, then went to three sets with Petra Cetkovska in the third round too, losing eight straight games at one point. But she's still a winner every time.
"I think I've learned how to be patient with myself," Stephens said. "Just learning how to compete, hanging in there and just battling, never giving up, just knowing that even the little things, like when you're down 30-0 in a game, 40-0 in a game, you can always come back. The game is never over until they're like, 'It's 3-2,' or whatever it is. I'm just really learning how to compete better out there."
After her breakthrough Australian season, Stephens hit a plateau - she won only two of her next nine matches and stayed put at No.17 in the world. But in the last month and a half she has won 11 of 14 matches, and is now primed to keep clawing up those rankings - how did she break out of her spell?
"I just kept believing in myself," she said. "I mean, it was a bad time. But just knowing that I am a good tennis player helped. I'm Top 20 in the world for a reason. I didn't all of a sudden snap my fingers and get good. I put in a lot of work, a lot of sweat, bad hair days, all that other stuff, to get where I was.
"I realized I couldn't just let that go to waste."
Bartoli is now through to the quarterfinals here for the third time in her career - she reached the final in 2007 (falling to Venus Williams) and the quarterfinals again in 2011 (falling to Sabine Lisicki).
"She hits really flat and hard, and goes for all her shots," Stephens said. "Like I said about Puig, it's tough playing someone who is going for it all the time. You have to adjust and do your best.
"It will definitely be tough. She plays really well on grass. She made the finals here. Obviously, she's a good player. I'm going to have to go out, do my best, compete, give 100%, and see how it goes."
Bartoli beat Stephens in their only previous meeting, in the early rounds of New Haven last summer. Stephens actually won nine games in a row in the middle of that match, taking the second set at love and building a 3-0 lead in the third - but Bartoli won six games in a row to clinch it, 61 06 63.