They now face a four day turn around before travelling to face Cardiff Blues and then head to Dublin for a re-run of last season's inaugural Grand Final against Leinster, who ran in three tries in an impressive 30-13 win over Ulster at Ravenhill.
Wales outside half Dan Biggar was the chief architect of the Ospreys' win as he scored 26 points in his side's fourth successive league win.
James Hook got the ball rolling with a first-half try as Biggar kept the scoreboard ticking over. But the floodgates opened in the second half with tries from Jonathan Thomas, Dan Biggar, Richard Fussell, Tommy Bowe, Paul James and Jamie Nutbrown.
The sell-out derby clash was postponed 24 hours after temperatures in Swansea plummeted. The game was rescheduled and the decision to go ahead paid off as the Ospreys romped home.
Sean Holley said: "Our players showed great endeavour and I'm pleased with both the result and score-line, but there are still things we have to work on.
"All around the park I thought we were fantastic. But we won't get too carried away with ourselves we are days away from another massive derby against the Cardiff Blues."
Rhys Priestland opened the scoring on three minutes but Biggar levelled the scores after nine minutes with a well-struck penalty. The outside-half added two further three-pointers to nudge the hosts into a narrow lead before Hook struck with a brilliant try.
Biggar failed with the touchline conversion but slotted two further penalties to give the Ospreys a 20-3 lead at the interval.
The Scarlets were dealt an early blow when Martin Roberts was sent to the sin-bin for mouthing off to the referee. Thomas scored two minutes later. Biggar missed the conversion but grabbed a try of his own five minutes later and added the extras.
But the Scarlets hit back thanks to a Gareth Maule try with Priestland converting. However, their comeback was short-lived with Fussell, Bowe and James grabbing tries while Sean Lamont was in the sin-bin.
Johnny Fa'amatuainu grabbed a late try for the visitors, but it was too little too late and the Ospreys had the last word when Nutbrown crossed in injury-time.
Scorers: Ospreys: Try - J Hook, J Thomas, D Biggar, R Fussell, T Bowe, P James, J Nutbrown; Con - D Biggar 3, J Hook 2; Pen - D Biggar 5
Scarlets: Try - G Maule, J Fa'amatuainu; Con - R Priestland 2; Pen - R Priestland