The Crimson Tide (18-8, 10-3 Southeastern Conference) finished a lackluster second half strong to claim its fourth straight victory. Alabama also closed the gap on SEC-leading Florida (11-2) after the fifth-ranked Gators lost to Missouri Tuesday night.
The Bulldogs (7-18, 2-11) cut a 13-point second-half deficit to four points before losing their 11th consecutive game to match the 1982 team for the second-longest skid in program history.
Cooper was 7-of-10 shooting for his first career double-double and Releford had five assists and five steals. Alabama's Trevor Lacey had 11 points before fouling out.
Jalen Steele returned from a three-game suspension for violating team rules to fuel the comeback attempt and score 15 points. Colin Borchert had 14 points and nine rebounds for Mississippi State, which committed 23 turnovers.
Steele scored 10 points in the second half and made 3 of 6 3-pointers. The rest of the Bulldogs missed their first 14 3-point attempts before they combined to hit three in the final minute.
Alabama led 31-20 at halftime and scored the first two points of the second half but couldn't put the Bulldogs away until the final minutes.
Mississippi State shot 12 of 22 (54.5 percent) in the second half. Alabama hit 21 of 27 free throws in the game.
Steele pulled up for a quick 3-pointer to make it 47-43 with 5:20 left after Mississippi State's Gavin Ware made a basket and free throw two minutes earlier.
Cooper answered with a 3-pointer for the Tide's first field goal in more than five minutes. Then Retin Obasohan had a steal and long pass to Lacey for a layup to push the lead back to 52-43 with 4:04 remaining.
Steele hit a couple of free throws after Alabama failed to score, but Releford got a steal and fast-break layup and then finished another fast break with a pass to Cooper for a dunk to end the threat.
Even with Steele's return, the Bulldogs remained with only six scholarship players. Coach Rick Ray suspended forward Roquez Johnson indefinitely on Monday for violating team rules.
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