Morata last hit the net for the league and Cup champions four months ago and has fallen out of favour with coach Massimiliano Allegri amid the spectacular rise of Argentinian Paulo Dybala.

But the former Real Madrid striker was given a rare start in the Cup version of the 'Derby d'Italia' -- as the Juve v Inter fixture is commonly known -- and reminded fans of his class with a performance that should have seen him bag a hat-trick.

Allegri told Rai Sport: "He had a good game, he broke the deadlock with a penalty that wasn't easy to score and then hit his second.

"Tonight, we gave ourselves a good chance of heading to the final, but let's not forget we still have 90 minutes to play at the San Siro."

Inter coach Roberto Mancini felt the scoreline flattered Juve, but the former Manchester City handler admitted they now face an uphill climb if they are to reach their first final since beating Palermo in 2011.

"They deserved to win but the scoreline didn't reflect the game," Mancini said.

"Up until their penalty, they hadn't threatened us and Samir Handanovic didn't have a save to make.

"It will be difficult for us to overcome the result but we have to go out and give it our best."

Morata stepped up to beat Inter 'keeper Handanovic from the spot nine minutes from the interval after Jeison Murillo felled Juan Cuadrado in the area to earn the first of two yellow cards on the night.

Morata hit a superb second just after the hour when he pounced on Felipe Melo's poor control of Patrice Evra's drive across goal, spinning away from his marker to fire into Handanovic's far top corner.

Morata admitted it had been "an important night for me", adding: "Thanks to all my team-mates for their encouragement and support."

Melo's mistake was too much for Mancini, who immediately replaced the Brazilian -- who only returned recently from a three-match ban -- with Marcelo Brozovic.

Brozovic injected some much needed punch into Inter's attack, but their hopes of reducing the arrears suffered a blow only minutes later when Murillo saw red for a second foul on Cuadrado.

It means Mancini will be without two defenders, Murillo and Joao Miranda, for the return leg on March 3.

Morata should have had his third on 78 minutes when Frenchman Paul Pogba curled in a delightful delivery for the Spaniard, but somehow Morata ballooned over from eight yards. 

Dybala, who has hit 12 Serie A goals this season, wrapped up the win only minutes after coming off the bench to replace Mario Mandzukic with a superb sweeping shot that went through Handanovic's legs six minutes from time.

Juve, who secured their first league and Cup double in 20 years last season, travel to Milan for the second leg as favourites for their second Cup trophy.

On Tuesday, fallen giants AC Milan edged third division side Alessandria 1-0 at Torino's Stadio Olimpico.