Trailing 13-3 midway through the second period, they eventually fought back to triumph 17-13 to place sixth in the GUINNESS PRO12 table.
"Making Europe means everything to the region," Owens told the South Wales Evening Post.
"It has been playing on my mind in the last couple of weeks. I didn't want to be the first captain not to achieve that.
"Treviso are no mugs, they pushed Leinster all the way in Dublin last week and were fighting for a European spot as well, and we knew it was going to be tough with the heat and conditions.
"We needed to cut our errors and we managed to do that to a certain extent in the second half.
"It wasn't pretty at times, but we dug deep and got the result."
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