Paulo Dybala continued his fine run of form by coming off the bench to score the third and Juan Cuadrado gave possibly his best performance since joining Juve at the start of the season as he repeatedly rampaged down the right flank.
Juventus created a flurry of early chances, Cuadrado firing narrowly wide of the post, and had a penalty appeal turned down when a Cuadrado free kick appeared to hit Gary Medel's hand.
They went ahead with a penalty when Cuadrado was fouled by fellow Colombian Jeison Murillo and Morata fired home from the spot in the 36th minute, ending a goal drought stretching back to Oct. 4.
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Morata struck again just after the hour, scoring from close range after Patrice Evra pulled the ball back into the goalmouth and Felipe Melo's attempted clearance went straight to him.
Inter's troubles continued when Murillo was given a second yellow card for barging into Cuadrado in the 72nd minute and they were left to play the rest of the match with 10 men.
Morata missed an excellent chance to complete his hat-trick when he was first to get to Paul Pogba's inswinging cross, but he fired over from six metres.
It was left to Dybala to complete the scoring, the Argentine stylishly side-footing home Kwadwo Asamoah's pass first-time from the edge of the area for his 15th goal in all competitions this season.
Roberto Mancini's Inter were left to contemplate a derby against AC Milan in Serie A on Sunday and a five-week wait for the return at the San Siro on March 3.
(Writing by Brian Homewood in Berne, editing by Ed Osmond)
