The 15-year-old was bidding to become the youngest player to reach a grand slam quarter-final for 15 years, and she was one set away when she took the opener on a packed Melbourne Arena.
But 14th seed Kenin had other ideas, eventually wearing down Gauff in a 6-7 (5) 6-3 6-0 victory.
The 21-year-old has had her own speedy rise, although without the attention of Gauff's, and will play in the last eight of a grand slam for the first time.
Kenin said: "It was such a tough match, she's such a tough player, all respect to her.
"It means a lot to me, I'm so happy to be through to the next round. I just tried to play my game, fight for every point and not focus on anything else."
Gauff's victory over defending champion Naomi Osaka was on paper certainly the best of her fledgling career but, so poor had the Japanese been, it was difficult to gauge how well the teenager had actually played.
Kenin, who knocked out Serena Williams at the French Open last summer, is a feisty competitor and a good tactician, and it was she who made the more solid start to open up a 4-2 lead.
But Gauff worked her way into the contest, gradually making her extra power count and forcing Kenin on to the back foot.
Gauff was hitting her second serve faster than Kenin was sending down her first – only two women have served faster than Gauff this tournament – and the teenager exploited some loose play from her opponent to level at 4-4.
Gauff had one set point at 5-6, which was saved well by Kenin, but two double faults in the tie-break were costly for the 14th seed and, although she saved three more set points, Gauff composed herself superbly to take the fourth.
The turning point came when Gauff double-faulted three times to drop serve in the fourth game of the second set and from there Kenin pulled away.