Sosa collected two hits in his first game for Alabama since the 2002 season, including his second career home run to cap a three-run third inning that saw the Tide take a 4-0 lead.
Alabama (1-0) scored its first run of the 2004 season in the bottom of the second inning when junior center fielder Matt Grooms drew a two-out walk and scored on a double down the left field line by Youngsville, La., native Rocky Scelfo.
The Crimson Tide added to its lead with a back-to-back home runs in the third inning to chase starter Matt Crockett.
After junior shortstop Mark Haske was hit by a pitch, junior third baseman Allen Rice smashed his first home run of the season over the right field wall. Sosa capped the inning with a line shot down the left field line to make it a 4-0 game.
Crockett (0-1) allowed four earned runs and four hits in 2.1 innings en route to the loss for UNO. The lefty also added four strikeouts.
Alabama starter and preseason All-America candidate Brent Carter threw six scoreless innings and retired 14 consecutive batters before UNO third baseman Jay Yaconetti reached on an error by Haske to open the seventh inning. Mike Nuckles followed with a walk and Carter was replaced by senior right-hander Seth Johnson.
UNO first baseman Dana Tosoni grounded into a double play and Johnson came within one pitch of stranding Yaconetti at third base before Kody Valverde broke up the shutout with a two-out single to left center field.
Carter (1-0) allowed one unearned run and two hits in six-plus innings with six strikeouts and one walk.
Johnson threw three shutout innings for his first career save for the Tide, allowing two hits and striking out two batters.
The win was Alabama's 11th straight opening day win, dating back to 3-0 win over the Privateers at the 1994 Winn Dixie Showdown at the Louisiana Superdome. Alabama has also 11 consecutive home openers, dating back to a double-header sweep of Austin Peay in 1995.
The series continues on Saturday at 4:30 p.m. with junior right-hander Thomas Diamond, a preseason All-America selection and potential first-round draft pick taking the mound for UNO. Alabama will send freshman lefty and Lake Charles native Wade LeBlanc, the 2003 National High School Co-Pitcher of the Year, top the mound for his collegiate debut.