Alcaraz was handed the reins at Levante on Tuesday after predecessor Jose Luis Mendilibar paid the price for a dismal start that yielded just five points from the club's opening eight league matches.
However, the appointment of Alcaraz failed to spark an immediate revival at the Estadio de Balaidos as Levante lost for the sixth time in the league this season and they remain in 19th position.
Larrivey scored in each half to put Celta into a commanding lead before a goalkeeping error from Diego Marino ensured substitute Alex Lopez also finished the match on the scoresheet as Celta climbed one spot to sixth.
Larrivey opened the scoring in the 27th minute with a smart finish from eight yards after a free-flowing move, although he appeared to be in an offside position when he received the ball.
Celta's lead was almost doubled shortly after the break when Santi Mina's cross narrowly evaded Nolito in the penalty area.
Levante's opportunities proved scarce, but Andreas Ivanschitz's thunderous strike from 25 yards beat Sergio Alvarez in the Celta goal but cannoned back of the crossbar.
That proved to be Levante's best chance and Celta moved 2-0 ahead in the 83rd minute when Larrivey nodded into the unguarded goal after Nolito's initial effort had been saved.
The points were then confirmed in the last minute of time when Lopez's effort from the edge of the area squirmed through the hands of Marino.
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