The home side suffered a heavy loss to Leicester in round 1, and their confidence took a dent when Wales wing Shane Williams opened the scoring for the visitors after just four minutes.
But that was to be the only time the visitors crossed the try line as the home side settled their nerves and into their rhythm.
A concerted period of pressure paid of, and it was another Welsh international, ASM's fly-half Stephen Jones who scored a 12th minute penalty. That was soon followed by a try from Alexandre Audebert, converted by Jones.
ASM had their noses in front, and despite a series of spirited Ospreys attacks, they held their nerve, and in the end ran in comfortable winners by 34 points to 14.