The Tigers (24-5, 14-2 Southeastern Conference) trailed by six points midway through the second half before taking over to maintain a share of the league lead.
LSU took a 64-59 lead on a layup by Mays with 3:19 left. Alabama (17-12, 8-8) cut it to one but then Riley Norris’ pass was intercepted by Tremont Waters to set up a big finish at the other end. Waters, who missed the past two games with an undisclosed illness, bounced the ball off the backboard and a trailing Mays caught it for a dunk.
Alabama made one final push, and John Petty’s 3-pointer with nine seconds left cut it to 72-69. Mays hit four straight free throws over the final 16 seconds to ice the win, including two after that basket.
Mays had 14 points over the final 9:37 to help the Tigers pull out the win.
LSU came into the game tied with No. 4 Kentucky and No. 7 Tennessee atop the league standings, with those teams playing each other.
Naz Reid had his second straight double-double with 12 points and 13 rebounds.
John Petty made five 3-pointers and led Alabama with 23 points. Donta Hall had 14 points and 10 rebounds, and Riley Norris finished with 13 points and nine boards. Herbert Jones, like Norris, delivered a strong performance off the bench with 11 points.
Alabama surged into the lead with 10 straight points a few minutes after coach Avery Johnson was called for a technical foul with 14:12 left. His assistants and son Avery Johnson Jr. had to corral him, but the Tide ultimately went ahead 47-41.
Waters entered five minutes into the game and hit a 3-pointer on his first attempt. He had five points and three assists in 24 minutes.
LSU: Led by 11 in first half and held on for its third straight victory. Hit 50 percent (17 of 34) from the field and made 10 of 11 free throws in the second half.
Alabama: Had a streak of seven straight home wins over Top 25 teams end. Made just 24 of 64 shots (37.5 percent).
LSU visits Florida on Wednesday night before facing Vanderbilt in its final two regular-season games.
Alabama hosts rival Auburn on Wednesday night.