The home side showed their attacking intent straight away, kicking for the corner instead of taking the three points, and they were rewarded when Duncan Weir spotted a gap and went over for the try.
Weir converted his own try, before Steve Shingler grabbed a penalty back for the Scarlets.
Naiyaravoro had a stunning try disallowed but Weir added a penalty for Glasgow just before half time.
Shingler kicked another Scarlets penalty after the break, but James Malcolm buried over after strong work from the forwards to extend Glasgow's lead.
Further impressive defence saw Glasgow turn the ball over twice in their own 22, and they made Scarlets pay after a pinpoint cross-kick from Finn Russell found Naiyaravoro for the four-try bonus point.
Naiyaravoro then grabbed his third try seven minutes from time after catching a lazy loop pass and cantering in from 60 metres out.
It is Glasgow's first win Europe this season and they move up to second in the pool, with Scarlets at the bottom with no wins.