The Irish led late but failed to make their free throws down the stretch and did not earn the win until Texas' A.J. Abrams' desperation shot from beyond half court bounced off the front of the rim.
The Irish took a 43-40 lead into halftime and appeared to run away from the Longhorns late in the second half.
Harangody knocked down a pair of free throws to give Notre Dame a 79-71 lead with a minute left and it appeared the game was going to be over. But Abrams knocked down a pair of three-pointers around two Jackson free throws to make the score 81-77 with 25 to play.
Zach Hillesland was fouled on the inbounds and missed two free throws, the second one an air ball, and Texas' Justin Mason drained a three to make it a one-point game with 10 seconds left. Harangody was fouled and he missed two free throws before Abrams' heave came up inches short.
Abrams finished with 23 points and Mason added 16 points and seven assists. Texas' Damion James chipped in with 11 points and 12 boards.
Harangody had the play of the game, picking up a loose ball as the shot clock was running down and heaving it in off the backboard from near midcourt to give the Irish a 66-60 lead with 7:45 to play.
Notre Dame improves to 4-0, while the Longhorns drop to 3-1.
The Irish beat Indiana 88-50 in the opening round and will face North Carolina, the nation's number one team, the Maui Championship on Wednesday night at 10 p.m. (EST). North Carolina beat Oregon 98-69 in the other semifinal. The Ducks will play the Longhorns in the consolation game on Wednesday.