It marked the first road win of the season for the Volunteers (9-8, 2/3 SEC). Mostella had a game-high 24 points while Punter had 23 points and five assists.
Mississippi State (7-9, 0-4) is off to a 0-4 start in SEC play for the first time since 1992-93.
Mississippi State was led by freshman Quinndary Weatherspoon, who made his first career start in the SEC. Weatherspoon had a career-best 23 points while Gavin Ware had 21 points and 10 rebounds for the Bulldogs.
It was Ware's fifth double-double of the season and the 17th double-double of his career. Ware has now scored in double digits in 16 straight games to open the season.
Tennessee shot 47 percent for the game and made 10 of 23 3-pointers and had just eight turnovers. Mississippi State shot 42.4 percent from the floor and just 5 of 19 beyond the arc.
The Bulldogs held a 40-31 rebounding advantage, including a 17-8 advantage on the offensive boards.
Tennessee held the second-half lead for the duration and led by as many as 11 points in the second half. A layup by Weatherspoon cut the Tennessee lead to 60-59 with 5:15 left but the Bulldogs would get no closer.
Tennessee used its perimeter shooting to take control in the first half. The Volunteers trailed 9-8 in the opening minutes but reeled off 13 unanswered points, including a trio of 3-pointers, to grab a 21-9 lead with 13:23 left in the first half.
Mississippi State managed to cut the deficit to 29-24 following a 3-pointer from I.J. Ready. But the Volunteers finished off the half with their sixth 3-pointer and held a 38-27 lead at halftime.
Tennessee shot 50 percent in the first half, including 6 of 14 beyond the arc. The Bulldogs managed just 35.7 shooting in the first half and were just 2 of 9 on 3-pointers.
Tennessee: Senior forward Armani Moore did not play Saturday after injuring his ankle on Wednesday in the Volunteers' loss at Georgia. Moore entered the weekend as the Volunteers' second-leading scorer at 12.4 points a game.
Mississippi State: On Saturday, the Bulldogs wore throw-back uniforms to honor an all-white 1963 squad that had to escape town in order to play an integrated Loyola (Illlinois) team in the Bulldogs' first-ever NCAA Tournament. Also on Saturday, the Bulldogs recognized the 1991 squad, a group that won the SEC regular-season championship and the program's first SEC title since that 1963 squad.
Tennessee hosts Vanderbilt on Wednesday.
Mississippi State travels to Florida on Tuesday.