The home side instantly seized the initiative and took the lead in the first minute through a Stephen Myler penalty goal.
A try from Samu Manoa extended Saints' lead on 12 minutes, with Myler scoring the conversion.
Ospreys hit back with two penalty goals in three minutes from Dan Biggar to get the away side back into the game.
Christian Day restored Saints' advantage on 26 minutes with the second try of the game, while Myler was accurate once again with the conversion.
Ospreys started the second half well and struck first on 44 minutes with a try from Biggar, who then stepped up to convert afterwards.
Ben Foden increased the gap between the teams with a try on 56 minutes as Saints began to get on top once again.
Six minutes later, Biggar closed the gap by three with a expertly-taken penalty goal.
Meyler replied with a penalty goal of his own minutes later to secure the win for the hosts.