Exeter responded with a try of their own towards the end of the first half, Will Carrick-Smith crashing over after good work by the forwards. Slade made no mistake with the conversion to make it 10-7.
This lead was extended just before the interval when Myles Dorrian went over following a flowing move for a second Exeter try.
Slade converted the try to make it 17-10 at half time, and throughout the second half kept the scoreboard ticking over for his side as they saw out the win.
Matthew Morgan had a late chance to salvage a losing bonus point for the Ospreys, but his penalty drifted wide as the Welsh side left empty handed.