Cucinotta hit his first blast in the opening frame, giving the Explorers a 2-0 advantage.
Tommy Toal collected a pair of hits in the win and Peyton Sorrels finished with a hit and an RBI.
For the Hawks, James McConnon reached base four times and collected two hits in the loss.
Colin Scanlon picked up the win for the Explorers, surrendering one run on five hits over five innings. Matt Holt picked up four strikeouts in three innings of relief.
The victory was La Salle's 20th of the season, matching a mark last reached in 2015.
The two clubs conclude their three-game set on Saturday afternoon at Smithson Field.
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