Lacey scored 14 points for the Crimson Tide (13-7, 5-2 Southeastern Conference), which held on for a fifth win in six games after losing 54-53 at Tennessee when he couldn't get off a final shot.
BJ Young hit a layup with 5 seconds left and then missed a 3-pointer at the buzzer for Arkansas (12-8, 3-4) after Randolph made two free throws.
Young, who led Arkansas with 14 points, managed to get a shot off with a behind-the-back dribble but overshot the rim. He scored 12 in the second half.
Marshawn Powell had 11 points for the Razorbacks before fouling out.
Young missed all four of his 3-point attempts and the Razorbacks were just 3 of 19. They are now 0-5 away from home and have totaled 110 points in two straight losses.
The Tide shot just 7 of 23 (30 percent) in the second half but also made 14 of 19 free throws (74 percent).
Arkansas forced a turnover with the full-court press and Rickey Scott collected the loose ball and a layup with 51 seconds left to cut Alabama's lead to 56-54. The Tide then worked most of the shot clock down before Andrew Steele missed a 3-pointer.
Lacey grabbed the offensive rebound, drew Powell's fifth foul and hit the second of two free throws with 17 seconds remaining for a 57-54 lead.
Young then drove to the basket instead of trying to tie it with a 3-pointer, and the Razorbacks called a timeout before quickly fouling Randolph. He made both with 4.2 seconds remaining.
The Razorbacks, who had trailed by as many as 11 points in the first half, went on a 17-5 run to take the lead following Releford's technical foul early in the second half.
Young hit two free throws for a 42-39 lead midway through the second half. Lacey answered with two free throws, Releford had a reverse layup and Tide freshman Devonta Pollard scored inside to get Alabama back on track.
The Tide, which came in last in the SEC in rebounding, controlled the boards 38-29.
Mike Brey is looking at former Notre Dame players to fill two assistant jobs that have opened up with the departures of Martin Ingelsby and Anthony Solomon.
Purdue's 2016-17 prospects got a jolt Wednesday when forward Caleb Swanigan announced he will return for his sophomore season.
Clemson's Jaron Blossomgame says he is withdrawing from the NBA draft and returning to school for a final season.