The painting called The Actor sustained a vertical tear of about six inches (15cm) in the lower right-hand corner.
But the damage did not affect the "focal point of the composition" and should be repaired for an exhibition later this year, the museum said.
The work from the Rose period was painted in the winter of 1904-1905.
The repair should be completed in time for the Picasso in the Metropolitan Museum of Art exhibition, which will feature some 250 works from the museum's collection and is due to open at the end of April.
The unusually large canvas, measuring 6ft by 4ft (1.8m by 1.2m) and which depicts an acrobat posed against an abstracted backdrop, was damaged on Friday.