James Arlidge kicked one and Shaun Connor converted two penalties for the visitors in reply.
"We've taken a bonus point away from arguably the best team in England," said Turner after the Pool Five defeat.
"We certainly thought we could beat Bath if we fronted-up for 80 minutes. It was our best defensive performance in the three years I've been at the club.
"I'm really proud of the players and the bonus point was well deserved. We are disappointed we didn't win but we needed that something extra which we didn't have.
"But we threw our bodies on the line and defended heroically against a fine side.
"No-one give us a prayer against Bath after their performance against Toulouse last week.
We got turned over too often and they came out and defended very well, as we expected
Bath boss Steve Meehan
"I knew we had to make 100 tackles to beat Bath and we did that, we hunted them down but we just couldn't get our kicking game in order so perhaps that was a fair result."
The Dragons' next Pool Five clash is at European powerhouses Toulouse but Turner urged caution as he said: "We are not going to win the European Cup.
"It's not about the here and now for the Dragons; we're building a side, we're building a future.
"We've a three-year plan and this is our first year - it's all about being competitive, which we were against Bath."
Turner did have a special word for veteran back-row Colin Charvis, saying: "You could do with five of him.
"He is the best in the business with his nose over the ball.
"Having said that I've been coming down here for years with different clubs. It's deja vu doing well and losing in the last 10 minutes."
Dragons flanker Joe Bearman, who was named man-of-the-match, "enjoyed" his day of non-stop defending against the 1998 Heineken Cup winners and last season's European Challenge Cup winners.
"It was a bonus point gained certainly," said Bearman.
"It was enjoyable but we were disappointed not to sneak a win at the final whistle.
"Tthe boys can be proud of their performance and any one of 20 of our lads could have won the man-of-the-match honour as we all stuck together as a team."
Bath coach Meehan believes the Dragons have the potential to shock pool rivals Toulouse when they meet the French aristocrats twice in December.
But reflecting on his own side's performance, he added: "We played some sound rugby but never really nailed it.
"We got turned over too often and they came out and defended very well, as we expected.
"We were not as mentally strong as we can be. It was difficult for the players but there were a couple of options up our sleeve that we had and didn't use.
"We will be upset with ourselves in that regard. What we said at half-time was that we hadn't gone out and achieved what we needed to do."
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