Scarlets fly-half Stephen Jones was once again criticised for his performance after Bowe intercepted his pass with just two minutes left to run in unopposed from 70m to seal the game.
The defeat, their 11th of the season, leaves the Scarlets in danger of missing out on Heineken Cup qualification.
Jonathan Davies scored a fine turnover try for the Scarlets, but Bowe also went over in the first half before his opportunist second sealed the result.
Ospreys head coach Sean Holley said: "I'm pretty relieved. I think it was a great derby for different reasons.
"The conditions meant it was quite tactical with a lot of kicking but full credit to the Scarlets, I thought they were excellent and pushed us all the way.
"With this bunch you're always confident you can get back in the game."