The Greek champions opened their best-of-five playoff series with the convincing win while 2010 champions
Nick Calathes starred as he scored 10 of 22 unanswered points for Pana in a run than spanned the first and second quarters and did the damage for the hosts.
Already comfortable, Pana then opened the second half with a series of three-pointers, taking it to a 30-point game, from where the result never looked in doubt.
Euroleague MVP Dimitris Diamantidis top-scored as he recorded 15 points and eight assists.
Barca were equally comfortable as Pete Mickeal collected 22 points and seven assists to help his side race out to a 28-14 lead in the second quarter.
Unics did manage to bounce back and at one point early in the fourth quarter had cut their opponents' lead to just five.
But Juan Carlos Navarro, Erazem Lorbek and Boni Ndong got things back on track for the hosts to secure a deserved win.
The action resumes on Wednesday with CSKA Moscow entertaining Gescrap BB and Olympiacos travelling to Montepaschi Siena.