They also scored four tries, to claim a crucial bonus point.
Neath’s secured the win a second half they dominated.
A scrappy first half saw the two sides in at 13-13. Five penalties for man-of-the-match Henson at the start of the second half was the run of points the gave the Ospreys the upper hand in the battle to establish supremacy.
Edinburgh ralllied and replyed with a try finsihed by left-wing Hugo Southwell.
The ospreys lost captain Scott Gibbs with a dead leg at the beginning of the second half, and stand-in skipper Barry Williams lead by example as the screw was turned.
It was from a line-out drive that the Ospreys earned their third try for the lock Andy Newman.
With the match all-but sealed the Ospreys went in search of a bonus point and it duly arrived when replacement centre James Storey broke free from a tackle to go over.
In the first half, Gibbs had crossed for a try along with outside-half Shaun Connor that had helped to cancel out a five-pointer from Derrick Lee after the Scottish side had led 10-3 after 15 minutes.
The Ospreys will now be hoping Gibbs recovers in time for the difficult trip to face Cardiff Blues at the Arms Park next Saturday.