Emanuel limited the Longhorns to four singles in his third win of the NCAA tournament and first career shutout.
Jacob Stallings hit a two-run single in the third inning and Ben Bunting finished a four-hit day with an RBI double in the ninth for the Tar Heels (51-15), who play Vanderbilt or Florida on Wednesday.
Texas (49-19) went two games and out for the second time in 25 CWS appearances since 1966 and for the fourth time in its record 34 trips to Omaha. The last time was in 2000.
Emanuel (9-1) walked one and struck out five. North Carolina's Robert Woodard pitched the last shutout here, blanking Clemson in 2006. The last freshman to do it was LSU's Brett Laxton in 1993 against Wichita State.
The Longhorns' offense struggled in their two CWS games, going three-up, three-down in 11 of 18 innings against Florida and North Carolina.
They twice ran themselves out of innings Monday, with Jonathan Walsh getting doubled off in the second and Mark Payton in the fourth.
North Carolina fans began chanting "Heels, Heels, Heels" after Bunting's double in the top of the ninth, and they applauded as Emanuel emerged from his dugout to start the bottom half.
He needed only nine pitches to finish off the Longhorns, getting Tant Shepherd to pop out to first, Mark Payton to ground out and Brandon Loy to fly out to right.
North Carolina, which left 16 runners on base in its first-round loss to Vanderbilt, got just enough offensive production to advance. The Tar Heels had nine hits against five pitchers.
Stallings came through with the bases loaded in the third, delivering a two-out single for a 2-0 lead.
Texas had been 8-1 in elimination games during the postseason, starting with the Big 12 tournament.
The Longhorns came to Omaha well-armed, but ace Taylor Jungmann gave them only 4 1-3 innings in the 8-4 loss to Florida and Monday starter Cole Green got hit hard and lasted just two-plus innings.
Green (8-4) matched his shortest start of the season after having gone at least five innings in each of his 12 starts dating to March 27.
It was Green's second straight rough outing in Omaha. In 2009, he lasted one inning as the starter in the 11-4 championship-game loss to LSU.