Morgan scored all of his side’s first-half points at the Sportsground as the Welsh Region extended their winning run in Galway.
The hosts took the lead on a blustery night on the West coast of Ireland, fly-half Miah Nikora kicking his side ahead with a penalty.
Morgan then levelled things up with his first penalty as the Ospreys came away with points after their first venture in the Connacht twenty-two.
That lifted the visiting Welsh region and they were next to score as well. Scrum-half Rhys Webb spotted a mismatch and darted between two Connacht forwards to tee Morgan up for his converted score under the posts.
Connacht through Nikora responded straight from the re-start to bring the scores to within four points at the break.
It took twenty second-half minutes for either side to add to their tally and it was prop Duncan Jones’ break up-field which set up the platform for replacement centre Parker to score.
