However the hosts fought back after the break and levelled things up thanks to Luciano Orquera's kicking and a penalty try.
But second-half scores from Liam Williams and No.8 Morgan saw Scarlets over the line and still in the hunt for some silverware come the end of the season, despite a late consolation from Aironi wing Giulio Toniolatti.
The Scarlets got off to a great start when Rhys Priestland knocked over a penalty after just three minutes.
But it could have been so much better as just moments later Ken Owens sent Nic Reynolds away in the corner but the centre knocked on with the tryline at his mercy.
After seven minutes Aironi were reduced to 14 men as fly-half Luciano Orquera was shown a yellow card but despite some constant Scarlets pressure the Italians levelled the scores with Tito Tebaldi's 15th-minute penalty.
Priestland and Orquera then traded three-point kicks before Scarlets flanker Edwards got the first try of the game when he finished off a full-length move in the right corner.
Fly-half Priestland added the extras and then a penalty three minutes before the break for a 16-6 half-time lead for the visitors, although they knew they couldn't relax as Josh Sole knocked on moments before the interval with the tryline gaping.
And it was the home side that got on the scoresheet first in the second-half, Scarlets captain Josh Turnbull sin-binned for not releasing and Orquera knocking over the resulting penalty.
The Scarlets were made to pay for Turnbull's ill-discipline as Aironi made their man advantage count after a scrum led to the award of a penalty try, Orquera converting to tie the game at 16-16.
Orquera then had the chance to put his side ahead for the first time but missed with a penalty on 55 minutes.
And just five minutes alter they found themselves behind again as Scarlets went from left to right, Sean Lamont creating space by going wide, allowing Williams to race away to score.
Priestland's conversion hit the post but breathing space did come in the form of Morgan's try, supporting a great break from Andy Fenby to go the final metre and touch down.
Fly-half Priestland again hit the left post with his conversion but the damage had been done and Scarlets saw out the last ten minutes for victory, despite Toniolatti going over at the death.
|Aironi Rugby Score Card|
|Scarlets Score Card|