The Scarlets opened the scoring in the fifth minute when a stray pass was scooped up by winger Jasmine Joyce and she ran in from 50 metres. Joanna Phillips added the conversion.
The Ospreys held the majority of the territory and outstanding defence by centre Jami Davies kept the Scarlets in their own half. Ospreys' first try came from a well time backs move with Ffion Bowen putting Charlie Murray into space and she scored in the corner.
With Davies adding a penalty the Ospreys held a slender 8-7 lead at the break.
The Scarlets had their backs to the wall in the early stages of the second half with Gravell and Edwards both sin binned but the Ospreys were unable to make their superior numbers pay - due in no small part to the heroic work of Sioned Harries at the breakdown.
Eventually the pressure told with the Ospreys moving the ball wide before a line-break saw hooker Carys Phillips scored on 66 minutes.
A tense finale was ensured after the Scarlets used their brute strength, driving the Ospreys back to their line to allow second row Ffion Jones to burst over the try-line.
Ospreys set up camp deep in the Scarlets 22 for the remainder of the match, but a determined Scarlets defence bravely held firm.
Ospreys team manager Adrian Howell paid tribute to his side who scraped by with the win.
"Although we have a lot to work on, the girls can be proud of their win against a good Scarlets squad. The Scarlets did well to slow us down and stopped us playing to our strengths.
"I feel it was the character of the squad that pulled us through to gain a win in a match that could have gone either way," he added.
Cardiff Blues were always in control against the Dragons, racing out to a 17-0 with tries by Rafiuke Taylor, Cath Salter and Philippa Tuttiett.
The Dragons rallied to score tries through Naomi Whybrow and Cerys Hale but Tuttiett put the game beyond reach after supporting a great break from scrum half Laura Prosser to score.
The Dragons had the final say when Whybrow scored her second try of the match but it was too little too late as the Blues recorded a 27-15 victory.
Blues head coach Nadine Griffiths was pleased with her side's performance; "As it was our first performance together I was especially pleased with how the team gelled together.
"Our defence for the last 10 minutes was tremendous especially as due to two yellow cards we were left us with only 13 players at one stage. It was a great team effort and the forwards provided some lovely ball from the set piece which allowed us to release our exciting back line."