Cristiano Ronaldo was only named on the bench with Andrea Pirlo keeping one eye on the midweek UEFA Champions League clash against FC Porto.
First blood fell to the visitors after 14 minutes when Joaquin Correa fired past Juve shot-stopper Wojciech Szczesny.
The home side managed to get back on level terms in the 39th minute, with Adrien Rabiot finding the back of the net with an impressive strike.
Juventus saw a bit more of the ball in the latter stages of the half, but they were unable to carve out another opening before the half-time break.
The dominance was carried over to the second half as the defending champions completed the turnaround in the 57th minute through Morata's finish after some good work by Federico Chiesa.
Morata completed his brace three minutes later after converting a penalty for his seventh league goal of the season.
Pirlo opted to introduce Ronaldo, Weston McKennie and Arthur Melo for the final 20 minutes, as Morata, Aaron Ramsey and Juan Cuadrado came off.
However, three goals were enough to seal three points for the Old Lady, who now seven points behind league leaders Inter Milan.