Madsen, who posted a course-record 66 on day one to set the pace, had fortified her position of authority with an eagle on her 2nd hole in round two.
At one stage, the Dane was three strokes clear of the chasing pack but a bogey on the 13th was followed by a double-bogey on the 15th and she slipped off the top with a 73.
Madsen said, ”I’m definitely happy to get through. It’s going to be fun from tomorrow. My highlight today was an eagle on the 2nd. I hit a pretty good drive down the left side, hit an 8-iron on the green, had like 150 yards and then I made a six-metre putt.”
Fouillet, who was handily placed after an opening 70, experienced a roller-coaster start to her second round with a bogey, an eagle and a double-bogey on her first three holes.
The 19-year-old, who is a student at the University of California, steadied the ship with a stream of sturdy pars before reeling off a quartet of birdies at the 9th, 10th, 16th and 17th in a 69 to coast into the match play stages on five-under 139.
“I keep saying that I'm kind of a diesel,” she said. “I’m always off to a slow start. It takes me a few holes to actually lock in. But I knew I was going to have a couple of opportunities on the back nine, so I stayed patient. I steadied myself up, and then just trusted the process. It worked pretty well.”
Céline Boutier was the last French winner of The Women’s Amateur Championship ten years ago.
“Céline is really inspiring,” added Fouillet. “The Women's Amateur Championship is obviously a great championship in a career, but I think it's important for me not to get on top of myself.
“What I'm trying to do right now is go step by step, match by match.”