The Dragons scored 33 unanswered points to come from 14-9 behind going into the final.
"The first half was pretty static and boring," Edwards said. "It was like a game of chess but I'm extremely pleased with our second-half performance.
"We have some talented individuals who began to express themselves. Their try spurred us on and it was important going back into the RaboDirect PRO12 next weekend that we finished with confidence and momentum."
Fly-half Steffan Jones broke the Dragons' scoring record for an individual match with 27 points and Edwards added: "He's a point-scoring machine and he and centre Jack Dixon, with his off-loading today, were particularly impressive."
Despite finishing pointless in their LV=Cup pool, London Welsh coach Lyn Jones was not too downbeat.
"We were focusing on next weekend's Aviva Premiership clash at Leicester," he said. "We needed to get some players some game time and get them back up to speed.
"We did create 15 try-scoring opportunities but failed to take any of them. We created so much but when the Dragons had their chances they took them and deserved victory."
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