Trailing 26-23 in the dying moments, the Welsh region looked set to suffer one of the greatest European upsets of all time.
However, with just seconds remaining, 19-year-old Morgan cracked over a stunning penalty from 40 metres to break Italian hearts and earn Sean Holley's men a draw which they scarcely deserved.
Kris Burton opened the scoring for Treviso with a penalty after four minutes and their New Zealand-born winger Brendan Williams was set for a long range try shortly after but play was brought back for a knock-on.
The Ospreys had their opportunities too, but looked slightly off the pace as they squandered the chances that came their way.
Justin Tipuric knocked on with the line at his mercy, while Dan Biggar missed two kickable penalties either side of a failed drop-goal attempt from former Wales captain Ryan Jones.
They finally awoke from their slumber midway through the half, however, when Tommy Bowe burst through a number of tackles to dot down for the opening try, which Biggar converted.
Treviso threatened occasionally from broken play, but it was left to the kickers to keep the scoreboard ticking over, Burton twice converting either side of a penalty from Biggar to leave Treviso trailing 10-9 at the interval.
If the Ospreys needed a wake-up call, they got it immediately after the break when Treviso loosehead prop Michele Rizzo stormed his way over the line for a converted try.
Biggar hit back with a penalty but it was Treviso who were now calling the shots, a fact borne out by a superb 47th-minute try by Benjamin de Jager, who weaved through and outfoxed Shane Williams close to the line.
Treviso now led 21-13 but shot themselves in the foot with an overthrown lineout, and Ryan Jones gratefully accepted for a converted try.
But the home side were still looking the more focused, and when Tommaso Benvenuti got outside the Ospreys covering defence, scrum-half Tobias Botes was on hand to score try number three for the hosts.
Benvenuti was then lucky to stay on the pitch after dumping Williams in the tackle. He received a yellow for the incident, with Biggar reducing the deficit to three points with the resulting penalty.
What had started out as a tedious affair had by now developed into quite some game.
Staring down the barrel, the Ospreys threw everything at Treviso and almost forged a try when Williams combined superbly with Bowe, but Williams just got back to intervene.
The Ospreys looked a spent force but had just enough energy to force one last penalty and Morgan belied his tender years to split the posts majestically and garner two hard-fought pool points for his side.
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