Lehi pitcher Jason James was dominant through five innings, limiting American Fork to one run on two hits, striking out eight.
But with one out in the bottom of the sixth, Buster Hardman's single the Cavemen's first hit since the first inning kickstarted a rally as Hagen Homstead reached third.
Madsen followed with a single to left field to plate Homstead and tie the game at 2. Colton Schaerrer, who in the first inning had the other American Fork RBI, came up next and drove in Hardman to put the Cavemen in front.
"I wasn't any more pressured than just a regular at-bat in region play," Schaerrer said.
Madsen also pitched three no-hit innings in relief, striking out five for the win.
"We were dodging bullets," Ingersoll said of the choice to bring Madsen in. "We just made a decision that if we were going to beat them we were going to beat them with, probably, one of our best."
The victory keeps American Fork in the winner's bracket. The Cavemen will play Lone Peak in the semifinals at Utah Valley University on Thursday at 4:30 p.m.
"We believe and that's all that matters," Madsen said. "If we believe then we're going as far as we want, but we believe."
Lone Peak 10, Bingham 4 • The Region 4 champion Knights led from the top of the first inning until the bottom of the sixth, when the Miners' Colin Gordon singled through the right side of the infield to tie the game at 4.
But Lone Peak's Colton Easton walked with the bases loaded in the seventh to force home the go-ahead run. The Knights then erupted for five more runs, including two from a fielders choice that resulted in an error, RBI singles from Keenan Kelshaw and Cole Gambill and a sacrifice fly by Eli Norman.
Bingham could only muster one hit off Lone Peak's Seth Corry during his four innings of work, but still had plenty of chances due to the 10 walks surrendered by the southpaw.