Lackey began the spurt with a three-point play with 2:10 remaining and also added a pair of free throws down the stretch to pace the Wildcats (4-0).
A switched from a man-to-man full court press to a zone press disrupted the Lobos (2-1) as Arizona made three steals during the final moments to seal the victory.
Felecity Willis turned her steal into a 3-pointer to put the Wildcats up 70-61.
New Mexico, which led 40-37 at halftime, closed the gap to 64-61 on a jump hook by Jordan Adams, who had 16 points and 13 rebounds in the game.
Willis and Tatum Brown each had 14 points for Arizona while Elizabeth Pickney had 10 points.
Miranda Sanchez scored 15 points and Sonya Bryant 12 for the Lobos.