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 Rodney 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. The foul shots made Noel's game-ending dunk meaningless.
"They fought and had great confidence and they played to win," Calipari said. "We played not to lose, which young guys do on the road at times."
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 Grant said were cramps in both calves.
He didn't return to the game after recording seven points, four assists and four steals to offset a 2-for-9 shooting night. Moussa Gueye gave Alabama a shot-blocking presence that nearly matched Noel. He swatted five away in 18 minutes, including one on Goodwin's drive with 15 seconds left that allowed the Tide to put it away at the line.
That block drew a high-five from Grant.
"That was potentially the game-winning block right there," the coach said. "I just wanted to acknowledge to him that was huge right there."
Kentucky's previous season low in points was 60 at Vanderbilt in a win 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.