It was the first victory for Harvard in the history of the tournament and comes at the end of a season which began with the team's co-captains being dismissed as part of an academic cheating scandal.
The Crimson will next meet Arizona in a West Region matchup after the Wildcats got past Belmont 81-64.
Harvard used patience on offense, usually working the ball to the end of the shot clock, and then swarmed on defense while limiting the Lobos to 37 percent shooting.
New Mexico, at No. 3, became the highest-seeded team to lose on the first full day of the tournament.
Laurent Rivard added 17 points for Harvard (19-9), hitting five 3-pointers while the Crimson made 52 percent from the field overall.
Rivard's 3-pointer with 6:17 to play put Harvard in front for good and the Crimson began to celebrate when Saunders hit two free throws with 17 seconds left that put them in front by six.
New Mexico (29-5), which won both the regular-season and tournament titles in the Mountain West Conference, was paced by Alex Kirk's 22 points.
The game brought together two coaches who were among the game's best players in their collegiate days -- Tommy Amaker of Harvard and Steve Alford of New Mexico. Amaker starred at Duke and Alford at Indiana.