On both occasions Treviso fly-half Kris Burton punished the Ospreys and he kicked four penalties in total, handing the Italian side a shock 12-6 victory at the Stadio Comunale di Monigo.
Trailing 12-3 at the break, the champions did come out fighting despite their numerical disadvantage and Dan Biggar reduced the arrears to six soon after the break - but the Welsh fly-half also missed a number of kicks at goal.
And despite dominating possession after the break, the Ospreys couldn't find a way through with Ashley Beck held up on the line, Biggar failing to find Hanno Dirksen who was well placed on the wing and Tom Issacs fumbling when in the corner.
"I think [our discipline] played a big part in the match but I think we still had our chances to win the match," said the New Zealander.
"Credit to Treviso, they took their chances and held on and we didn't take ours after the break which was disappointing.
"We gave them their chances though and they did what they do and kept making it 3, 6, 9 and that was because of our indiscipline."
Meanwhile, Treviso head coach Franco Smith believes his side prevailed because they fought fire with fire.
"I think we kept our concentration right to the end," said Smith. "We didn't have much of the ball in the second half but we were very physical.
"We knew that they would be a very physical side and you have to fight that with physicality and I think we did that."
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