In the end the two-time Heineken Cup winners' experience proved the difference but the Ospreys hung on for a vital losing bonus-point.
Bowe is now hoping to go a step further and claim a win on home soil on Saturday.
"I've got very mixed feelings about the match," said Bowe.
"We are a team that really wants to do well in the Heineken Cup and we believe that we have the ability to really succeed in the competition.
"If you are going to do that then you've got to thrive on coming to places like Thomond Park, you have to test yourself and you have to believe in your abilities.
"We believed that we could get the win so obviously we are disappointed that we haven't achieved that.
"We have to be happy in a way that we got the bonus point, there are a lot of good teams who have failed to do that. Just look at Toulon when they came here recently.
"It's a massive game coming up in the return leg next week. I suppose it's only after that match, possibly not even until January, will we really be able to gauge how important that point is going to be, along with the fact that we stopped them getting a bonus.
"At different stages it looked as though we might be able to kick on and win. It was disappointing to concede a penalty right on half-time which meant we went in behind. To go in ahead would have been massive for us.
"But we did show glimpses of what we could do, and maybe we should have tightened up our game after we got our noses in front again in the second half."
"It's frustrating that we didn't come out with the win but at the same time we've got a bonus point and we can now look forward to a massive game and a fantastic rugby occasion on Saturday.
"It's only half-time in this tie and there's a huge opportunity for us next week in front of our own fans.
"We'll try to put things straight and we'll be looking to play a bit of rugby to get our supporters excited.
"We've got a very proud record in Europe at the Liberty Stadium and we want to keep that going.
"We're going to have our biggest crowd of the season there, it's going to be another fantastic game of rugby, you can't help but be excited about it.
"Our fans are going to Blackout the stadium, we need them to really generate a special atmosphere like we saw at times on Sunday.
"It's a fantastic competition, we obviously believe that we can progress through the pool stage despite the disappointment we are feeling now, and to do that, everyone has to play their part next Saturday, on and off the field."