DTH van der Merwe scored two tries as Gregor Townsend gained his first win as Glasgow Warriors boss.
The Warriors defeated the defending champions, to condemn the Ospreys to their third successive home defeat.
Only three minutes were on the clock when van der Merwe sprinted 70 metres with the ball to score the game’s opening try.
Having lost to Ulster and Scarlets, Glasgow were determined to extend their lead and did so through a Duncan Weir penalty.
Chris Cusiter was forced off with an arm injury but it couldn’t take the shine off an impressive victory.
A fumble from Rhys Webb allowed Chris Fusaro to pounce on the ball and run into the corner. Weir’s conversion saw Glasgow take a 15-0 lead.
Webb’s nightmare continued as he saw yellow for a professional foul and Weir took advantage to make it 18-0.
Both sides were reduced to 14 when Eddie was sin-binned and Ospreys got their first score when Evans went over the line.
But van der Merwe was determined to steal the headlines and he powered over the line for his second try of the match.
Ruaridh Jackson rounded off the win with a penalty late on.