Kyle Brindza kicked three second-half field goals, the go-ahead kick from 25 yards with 3:03 remaining. He tied a Notre Dame record with a 53-yarder that matched the longest in a college game at the $1.2 billion home of the Dallas Cowboys.
It was the second win in Texas in the five-year Shamrock run of home games on the road for the Irish (4-2), starting with a victory over Washington State in San Antonio in 2009.
Taylor Kelly threw for 362 yards and three scores for the Sun Devils (3-2), who missed on a chance to become the first team to beat Southern California and Notre Dame in consecutive weeks.
T.J. Jones had eight catches for 135 yards, including a pair of first-down catches on the drive to the winning field goal. Cam McDaniel, went to school in nearby Coppell, had four straight carries for 32 yards to get Notre Dame in field goal range.
The Irish clinched the win when Dan Fox, who recovered a fumble to set up a touchdown, returned Kelly's second interception 14 yards for a touchdown with 1:08 remaining.
The Sun Devils pulled within 37-24 on Kelly's third TD -- a 16-yarder to Marion Grice with 11 seconds left -- but Jones recovered an onside kick.
Notre Dame, one of the best in the country at rushing the passer last year but statistically one of the worst this year, found its form. The Irish pressured Kelly throughout, and Prince Shembo had a key sack on the first play after Notre Dame went ahead.
The Irish led 24-13 after Rees' 21-yard touchdown on a jump ball to 6-foot-7 tight end Troy Niklas, but the Sun Devils got back in it with a 37-yard interception return by Osahon Irabor and Kelly's 21-yard scoring toss to De'Marieya Nelson that made it 27-all.
Notre Dame's first regular-season game in the Dallas area since 1958 was technically a home game, and had that feel.
Both end zones were labeled "Notre Dame," with the Texas-themed Shamrock Series logo at midfield. The huge video board constantly showed smiling Notre Dame fans, and any sign of ASU yellow or red was by accident.
Rees finally gave the green-clad bunch something else to get excited about with two touchdown passes in the last 3:35 of the first half.
Ben Koyack scored his first career touchdown for Notre Dame when he caught a pass from Rees on the sideline with nobody around him and ran most of the 19 yards untouched for a 7-6 lead.
Kelly gave ASU the lead back when he took a snap with the play clock at zero and hit Jaelen Strong in stride behind the defense for a 36-yard touchdown on fourth-and-4. Strong had eight catches for 136 yards.
Rees, who had 156 yards passing in the second quarter, put the Irish in scoring range again with a 35-yard pass to Jones, then found Jones in the end zone from 8 yards for a 14-13 lead with 10 seconds left in the first half.
Arizona State's first game of any kind in the Dallas area since 1975 against TCU was a homecoming for coach Todd Graham, who grew up in the Dallas suburb of Mesquite and coached high school football in the area.