A paid crowd of 4,415 fans at Sewell-Thomas Stadium watched the game Friday night.
Carter (2-0) pitched into the seventh inning for the second week in a row and allowed three runs on six hits for his second win of the season. The preseason All-America candidate from Americus, Ga., added five strikeouts and walked three batters.
Rice scored two runs and added two hits to extend his hitting streak to five games. Bush also added two hits, scored one run and drove one run as he extended his hitting streak to three straight games.
Rice and Bush have combined for 12 hits in the last three games for Alabama.
It was Rice and Bush that put the Crimson Tide on the board in the bottom of the first inning. Rice doubled into the left field corner and scored on Evan Bush's two-out single to center.
The Crimson Tide added three more runs in the bottom of the fourth inning to take a 4-0 lead.
Bush led off with a double to left-center field and moved to third on a Rocky Scelfo's ground out. Michael McCallister followed with a sacrifice fly to deep center, scoring Bush from third.
Senior catcher Nick Richardson followed with a single and moved to second base when freshman Greg Paiml reached on a throwing error by UC shortstop Drew Saylor. Freshman Brandon Belcher followed with an infield single to load the bases.
UC starter Justin Minges then balked home Richardson from third to make it a 3-0 game. After the balk, Rice was intentionally walked to re-load the bases and then Sosa followed with an infield single to score Paiml from third. Junior Zac Welch grounded out to end the inning.
Cincinnati scored its first run of the game off Carter in to the top of the fifth inning when Erik Eitel drew a two-out walk and scored on Saylor's single up the middle.
The Bearcats chased Carter in the seventh inning with two runs to cut the deficit to 4-3.
Aaron Moll singled to leadoff the inning and then Carter walked the nine-hole hitter Mark Muscenti before leaving the game.
Senior Seth Johnson, making his fourth appearance of the season out the pen, got two quick ground outs, with the second grounder by Brian Beltz chasing Moll home from third base. Saylor followed with a two-out single up the middle to score Muscenti and cut the UA lead to 4-3.
Alabama added a run in the bottom of the seventh inning when Rice reached on a two-base error by UC first baseman Jack Nelson and scored on Sosa's second RBI single of the game, giving the Crimson Tide a 5-3 lead.
The Crimson added its final run in the bottom of the eighth inning when Scelfo led off with a double to right-center field and scored on sophomore catcher Bo Hannah's one-out single to center. The RBI single was the first career hit and RBI for the Tuscalooosa native.
Johnson was solid again out of the pen for Alabama, notching his second save of the season with three scoreless innings on the mound. The Sandy, Utah native retired nine of the 11 batters he faced and allowed two hits and struck out three.
Minges (0-1) went the distance for the Bearcats and allowed six run on 10 hits in the game. The right-hander added three walks and three strikeouts in the loss.
The series continues on Saturday with a 4 p.m. game at the Joe. Freshman left-hander Wade LeBlanc is the scheduled starter for Alabama.