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Juventus had lost two and drawn two of their four Serie A matches.
To win their first Serie A win against Spezia, Juventus fought back in the second half.
Six minutes after Federico Chiesa's equalizer, Matthijs de Ligt, centre-back, scored from a corner.
After Moise Kean's goal in the bottom corner, Juventus took a 28-minute lead. Spezia had made their own comeback.
Victory lifts Massimiliano Allegri's team up to 12th place in the table.
After four matches, which saw Juventus finish in the bottom three with two points, Allegri described the match as a "relegation 6-pointer".
After failing to win their first four matches, Kean's first goal for Juventus since joining Everton in august temporarily lifted some of the pressure.
Five minutes later, Gyasi equalized with a looping strike. After a few good saves from Jeroen Zoet, Paolo Dybala's goalkeeper, Antiste's deflected shot looked set to prolong Juve's poor start.
Chiesa was rewarded for his perseverance as the ball broke kindly for him goal. De Ligt scored with 18 minutes left before a nervous-looking Juventus managed to hold on for the valuable first three points.