The ninth-seeded Crimson Tide (32-24) faces regular-season champion Mississippi State Wednesday. No. 8 seed Kentucky (32-25) awaits its NCAA regional fate.
Alabama led 3-2 going into the ninth before Chandler Taylor drew a two-out walk and Haynie brought him home with a shot over the left field fence.
The Tide hadn't scored since a three-run second. Alabama had gotten a bases-loaded walk by Daniel Cucjen and Hunter Webb's two-run single.
Evan White hit a solo home run in the fourth and Riley Mahan doubled home another run in the seventh for the Wildcats.
Nick Eicholtz (4-3) allowed three hits and two runs in six innings for Alabama. Thomas Burrows retired the final four batters for his 12th save.