The Italy international, who joined the club from Roma on a four-year deal at the end of August, was finally able to end his drought in the 2-0 win over Crystal Palace.
The 27-year-old collected a pass from Adam Lallana before firing home from the edge of the box two minutes into the second half, with Rickie Lambert adding the second shortly after.
And Osvaldo could not hide his joy after opening his Premier League account.
"I was waiting for this goal since I started to play and I'm doubly happy because of the goal, and also because we got the victory," he told Southampton's official website.
"I was happy with this and as I have said before, I will be doing everything to score to help us in our next matches."
He also reserved special praise for Southampton's supporters, adding: "Not only for the goal, but since I arrived here, the fans have been fantastic with me.
"They show me every time their love for me and I wanted to start paying back with goals if possible, but if not, then with attitude on the pitch."
Saturday's victory lifted Mauricio Pochettino's men up to fourth in the table.
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