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Osimhen's cushioned volley put the visitors ahead after 10 minutes at the Stadio Luigi Ferraris before two good saves from David Ospina denied Samp an equaliser.
"For me it is important to start well, this gives me the confidence to score more goals," Osimhen told DAZN after the match. "Spalletti has helped me with the faith he has given me, and I hope to repay that. I am honoured to have a coach like him."
Osimhen, who had enough opportunities to score a hat trick, took his tally to five goals in five games in all competitions as the Partenopei continued to make an early statement about their title credentials.
Lazio are seventh in the standings with eight points, with Torino one point behind in 10th and Sampdoria 13th on five points.