Junior Drew Zimmerman started the top of the seventh with a double the opposite way to the corner in right field. Grace's sacrifice fly then moved Zimmerman to third and the next batter, Tyler James, then delivered the run with his own sacrifice fly to right.
With two outs, the Pioneers got extra life when a fly to right was misplayed by Cottonwood (23-5). Then the floodgates burst as Devin Smith drove a double to deep left-center and Kelton Higgins sent him home with a single.
James then socked the only home run of the game, sending a ball over the left-field fence.
"It was a curve ball. I knew [Cottonwood pitcher Brayden Bonner] he would throw it again," James said. "It just didn't break as much as it did the first time."
Colts' senior Wil Jensen started the game and was staked to a quick 3-0 lead as his counterpart, James, had control problems at the outset. Jensen and Cam Daley each singled in the bottom of the first, but that accompanied three walks and a Lehi infield error.
The Pioneers got two back in the third with an RBI double by Tyler James and a run-scoring single by Broc Hansen before Jensen worked his way out of a bases-loaded jam with a pair of strikeouts.
But Lehi eventually tied the game in the fifth when Devin Smith singled home Hansen.
Lehi, the third seed out of 4-5A, will play region rival American Fork in a quarterfinal game next Tuesday. Cottowood, which won Region 3, faces an elimination game against Jordan on Monday.
"The top of the seventh, our lead guy gets a double and our guys start believing," Lehi coach Jason Ingersoll said. "This Cottonwood team — they're a program. It's taking us a few years to become a program and this Cottonwood team has been ranked number one all year long.
"I'm really proud of our guys," he added. "They hung in there. Those first two innings were a little rocky, things didn't go our way. But James settled down and that was exciting to see, because we knew what he could bring."