Jermaine Marshall (29 points) gave the Sun Devils a 67-66 lead with 14.7 seconds left and on the ensuing possession Arizona's T.J. McConnell drove the lane and put up a shot that would have put the Wildcats ahead.
But Bachynski, the nation's leading shot-blocker, swatted it away. Jahil Carson picked up the loose ball, ran the length of the court and finished with a slam dunk to seal the victory for ASU (19-6, 8-4 Pac-12), which logged its first home win over a Top 5 team since Feb. 20, 1992.
Bachynski had eight blocks overall to go with 13 points and Carson scored 17.
McConnell finished with 17 points for Arizona (23-2, 10-2), which dropped a second in four games since a 21-0 season start.
Nick Johnson supplied 14 points and nine rebounds.