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Pizarro atoned for his error shortly afterwards, restoring Fiorentina's lead with a powerful strike into the top corner with the keeper out of position.
Mario Gomez came within inches of creating a bit of breathing space for his side when he curled a Rossi lay-off towards the bottom corner, but his shot came back off the foot of the post.
Fiorentina almost made it 3-1 in freak circumstances in the second half when Manuel Pasqual's cross threatened to go straight in, but Andujar just about recovered in time to keep it out.
La Viola could not extend their lead further in the closing stages but did manage to hold out for a win in their opening match of the new season.