Burns sank four-foot birdie putts at 16 and 17 after Scheffler birdied to win 15 in a 2&1 victory over Japan's Hideki Matsuyama and Ryo Hisatsune in the final early match at Medinah, leaving the Internationals leading by 8.5 to 5.5.
"I felt like if I could really crunch a seven-iron it was the perfect number and fortunately I was able to hit a great shot," Burns said of his stellar approach shots.
The hosts, swept in Friday's foursomes to fall behind 7-3, hoped to grab the lead in four afternoon foursomes (alternate shot) matches.
The Americans, who lead the all-time rivalry 13-1-1, seek an 11th consecutive victory with each side needing 15.5 points after Sunday's concluding singles matches to hoist the trophy.
The United States, with eight of the world's 11 top-ranked golfers, grabbed an early lead in all four Saturday four-balls (best-ball) but saw three of those leads erased.
South Koreans Kim Si-woo and Tom Kim improved to 3-0 this week by beating Wyndham Clark and Collin Morikawa 2&1.
Morikawa birdied to win the second hole and sank a 20-foot birdie putt at the par-three seventh for a 2-up lead.
Kim Si-woo birdied from just inside eight feet to win the par-three ninth and sank a 10-footer to birdie 11 and level the match.
Tom Kim put a finger to his lips to shush the US crowd after his 11-foot birdie tied the 12th hole.
Kim Si-woo sank a birdie putt from just beyond 12 feet at 16, then made a 16-foot birdie putt to win 17 and seal victory.
"Those last two were tricky putts," Kim Si-woo said. "My read, I would have missed, but (Tom) gave me so many good reads, so I just trust him, putt the line and trust myself."
'I'm gutted'
Australian Min Woo Lee and South Korean Im Sung-jae won two of the last four holes to tie Justin Thomas and Jackson Koivun.
Koivun won the first hole with a birdie putt from just inside eight feet and birdied the par-five sixth from just inside five feet for a 2-up US lead.
The 21-year-old US rookie added birdies to win the eighth and 11th and restore the 2-up margin, but Lee made a 16-foot birdie putt to win the 15th, and Im won the 16th with a par to level the match.
"I'm gutted," Thomas said. "I feel like I let (Koivun) down today and the team. I'm disappointed."
Cameron Young and Chris Gotterup won the first four holes and never trailed in beating Canada's Nick Taylor and Corey Conners 2&1.
Taylor won the eighth and Conners the 10th with birdies. Taylor and Gotterup each sank birdie putts from about 50 feet to tie the 14th, but Gotterup sank a 34-foot birdie putt at 15 and birdied the 17th from 33 feet to win the match.
"Felt like I was putting good all day, just burning edges, and finally got them to go," Gotterup said. "It was a good time."
Scheffler's tap-in birdie won the par-five fifth, but Matsuyama answered with a 35-foot birdie putt at the par-three seventh.
Burns won the ninth with a four-foot birdie putt; Matsuyama birdied the 10th from 34 feet, and his 10-foot birdie putt won the par-three 13th.
Scheffler sank a 21-foot birdie putt at 15 to ignite the rally, setting up Burns's heroics.