With the advantage after just 15 minutes, the rest of the first half was all in the Ospreys favour with good runs from Joe Thomas and Stuart Nicholas seeing them take the ball towards the try line before being bundled into touch.
But as much of the ball as they saw, the two sides went into the break with the score remaining at 6-0.
Both sides started the second half strongly but it was the Ospreys who took advantage when a ball from the scrum was passed along the line to Thomas to score. Price missed the conversion but at 11-0, the visitors were now in control.
To add insult to the Scarlets' injury, they were then further hampered when prop Richard Bassett was yellow carded.
Bizarrely though, being down to 14 saw the home side then score their first points of the day, Jones intercepting to race clear for a score he himself converted.
Buoyed by the try, the Scarlets kept up the pressure but some more spectacular tackles, particularly from Ospreys second row Mathew Dodd, kept them from getting back into the game. However, had a drop goal effort and penalty been successful, it may well have been a different story.
Those misses proved to be crucial and in the final 10 minutes, a Morgan Jones try, converted by fellow replacement James Davies secured the win for the Ospreys.
Their head coach Shaun Connor said: "Obviously we still have a lot to work on but tonight it is a massively positive start.
"The game was a little bit inaccurate but our physicality and contact area work was superb and that's what won it for us.
"We struggled a lot last year with our contact area so we have really focused on that over the summer and that paid dividends for us tonight.
"It was good to see Harrison Walsh back on the field for us, Joe Thomas looked sharp after his return from an operation while Joe Gage also played well at full back.
"Generally I thought the whole pack played really well."
In the other game to take place, a double-try burst from young hooker Liam Belcher ensured the Cardiff Blues Under 18 secured a 22-10 victory over the Dragons at Cardiff Arms Park.
Captain Belcher crossed twice during the first half as a commanding opening 40 minutes all but secured a win for the Blues.
Prop Rob Lewis got the home side off to a flying start before Dragons fly half Arwel Robson reduced the advantage with a penalty.
Belcher's brace in the space of just two first half minutes, both converted by outside-half Tyler Williams, saw the Blues storm into a 19-3 lead and a further penalty from Williams on the stroke of half-time put the hosts clear.
The Dragons scored the only points of the second half with a try from Leon Brown, converted by Robson, but there was to be no comeback.