Elliott Jones and Harley Thompson crossed for the Blues who had Diggy Bird to thank for their remaining points - two conversions and three penalties.
The Ospreys, who led 13-9 at the interval courtesy of a Ben Thomas try and a conversion and two penalties from Tom Priday, were outplayed in the second half and eventually went down 23-13.
First half tries from Jack Roberts, Aled Davies and Tomos Ellis Williams put the Scarlets on course at Stradey Park and Kirby Myhill made sure of a bonus point in his side's 38-18 success. James Russell added three conversions and a penalty and Dale Ford three penalties.
Lewis Roblin with two penalties and a conversion, finished with eight points whilst there were tries for Aledc Cooper and Nicky Williams.