The hosts almost made a perfect start to the match when Carlos Tevez got on the end of a header back into the box from a corner.
However, the Argentine was denied by Samir Handanovic from close range, before the keeper made another fine stop to deny the striker again from the rebound.
Handanovic could do nothing about Juve's opener in the 15th minute, though, as Pirlo showed his class with a delightful ball to Lichtsteiner, who nodded home into the far corner.
The Italian champions had a chance to double their lead in the 39th minute when Paul Pogba led a quick break before finding Tevez with a pass, but he could only curl an effort into the arms of the keeper.
Inter's first real chance came just a minute later when Rodrigo Palacio was played through on goal, only to lift his shot high over the bar.
Juventus responded with another chance of their own shortly afterwards, but once again Handanovic was alert enough to deny the home side, this time pushing Kwadwo Asamoah's shot away.