The hosts took the lead inside the opening three minutes as Arturo Vidal lashed in an unstoppable left-footed drive after the ball had broken to him in the area.
Vidal then made space for himself moments later to double his side's lead with a powerful effort into the far corner.
Juve's third of the evening came in the 26th minute when Roma goalkeeper Maarten Stekelenburg was sent off for bringing down Claudio Marchisio in the box. Despite substitute keeper Gianluca Curci saving from Andrea Pirlo's spot kick, the midfielder was on hand to convert the rebound.
The win was sealed 10 minutes after half time as Marchisio curled the ball past Curci following Mirko Vucinic's neat backheel.