Edwards saw his troops pick up their first league win since September thanks to second-half tries from Martyn Thomas and Lewis Robling.
But they nearly had the victory snatched away from them in one of the last plays of the game when fly-half Dan Biggar hit the post with an attempted drop goal, leaving Edwards a relieved man.
"I just think we completed good bouts of play and put some real pressure on the Ospreys and to get two tries," he said.
"My heart was in my mouth for the drop goal and it would have been really devastating for the boys. We had to go over the edge for the boys.
"We have just to capitalise week on week and it is a performance business and the results haven't gone our way recently but hopefully we can build some momentum."
Dragons fly-half Jason Tovey also contributed 11 points with the boot although he later left the field with a suspected knee injury.
But the 22-year-old says the pain was worthwhile and hailed the never-say-die team spirit at Rodney Parade.
"One of the main things about the club is all working for each other and you can see it in the boys at the end," he said.
"That's what we needed was that big win so hopefully we can really push on from here.
"I think there was a bit more grit and determination. We came out firing in the second half firing which we have not done in a long time."
While he was not on the scoresheet, No.8 Toby Faletau was once again at his rampaging best making hard yards and tackling with a fury.
That was enough for him to pick up his second successive RaboDirect PRO12 man-of-the-match award and Faletau was quick to echo Edwards sentiments.
"It is great we finally got a result and that can hopefully kickstart our season," he said. "When the drop goal hit the post I was just relieved. If the boys keep working hard we will take it from there."
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