And with several regulars missing with Wales as they prepare to take on Australia in Shane Williams last international appearance at the Millennium Stadium next weekend, Morgan again showed his worth for Scott Johnson's side.
The young outside half scored the only try of the first half, as well as adding seven points with the boot to send the Ospreys back to the top of the RaboDirect PRO12 win after Leinster's earlier win.
Connacht remain in ninth as they suffered a second successive home defeat to Welsh opposition after their 26-20 loss to Cardiff in their last home league game.
However the home side made the brighter start as Miah Nikora bounced back from missing an early penalty to give Connacht the lead after a quarter of an hour.
But Matthew Morgan brought his side level with a penalty of his own three minutes later.
And Morgan's day got even better on 32 minutes as he went over for the first try of the day which he converted himself to give the Ospreys a 10-3 lead.
Nikora managed to pull one penalty back for the home side before the break as they went in 10-6 down.
But Connacht were not able to take advantage after Ian Gough was sent to the bin in the second half, and the visitors stretched clear on the hour.
Duncan Jones, on his 150th appearance for the Ospreys collected the ball off the back of a line and went on a rampaging run before the ball found its way to Richard Fussell who fed replacement Sonny Parker to stroll over.
Morgan added the extras to extend the Ospreys lead to 17-6 with 20 minutes remaining.
Eric Elwood's men kept on pressing but a lack of accuracy in the visitors' 22 cost them any chance of a comeback.
In fact the Ospreys should have added to their lead as Morgan scooped up a knock on in his own 22, streaking down the pitch before finding Hanno Dirksen but the replacement winger couldn't break through.
The result was the perfect way for the Ospreys to warm up for next week's huge clash with reigning champions Munster.
|Connacht Rugby Score Card|
|Ospreys Score Card|