The Tide (12-6, 4-1 Southeastern Conference) held Kentucky (12-6, 3-2) to a season-low scoring total, 22 points below its average.
Levi Randolph and Trevor Releford each hit two free throws in the final 13 seconds to secure the victory for Alabama, which had lost seven of the previous eight meetings.
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Releford had 13 points and Randolph added 11 for the Tide. Jacobs came in averaging 5.8 points a game.
Kyle Wiltjer led Kentucky -- which had won four of five games -- with 14 points and seven rebounds. Nerlens Noel had eight points, 13 rebounds and for the second successive game blocked seven shots.
The Wildcats shot 8 of 27 (30 percent) in the second half, managing only 22 points.
Alabama and Kentucky traded timeouts after Wiltjer missed a potential tying free throw with 1:20 left and the Tide leading 51-50.
Jacobs, who came off the bench, scored on a putback with 51 seconds left for a 53-50 lead and then Julian Mays and Archie Goodwin missed shots on the other end. Releford dished to a driving Cooper, who was fouled and made one of two free throws with 28 seconds to play.
The ball went out of bounds off Kentucky after the miss.
Alabama's Trevor Lacey turned in a key play for a 48-44 lead with 4:26 left. Alex Poythress fouled out with six points, seven below his average, trying to stop his drive to the basket, but Lacey lay on his back for a couple of minutes while trainers attended to what appeared to be leg cramps.
He didn't return to the game after recording seven points, four assists and four steals despite a 2-of-9 shooting night. Moussa Gueye gave Alabama a shot-blocking presence that nearly matched Noel. He swatted five away in 18 minutes.
Kentucky's previous season low in points was 60 points at Vanderbilt in a victory on Jan. 10. A drought after halftime helped ensure the Wildcats fell short of that.
Alabama opened the second half on an 11-1 run to turn a nine-point deficit into a 35-34 lead on Lacey's two free throws with 13:29 left. The Wildcats missed their first eight shots and final three before halftime.
Ryan Harrow ended that drought with an open layup set up by Noel's block on a drive by Lacey. Harrow, who had scored in double digits for eight consecutive games, was 3 for 12 shooting and had six points.
The Wildcats didn't hit a 3-pointer for nearly 13 minutes to open the game, then made three quick ones. Mays hit two and Wiltjer followed up with another 3-pointer during a 17-6 run that stretched Kentucky's lead to 11 points late in the first half.
Randolph's shot in the final seconds didn't go in and the Wildcats took a 33-24 lead into halftime.