The Wales Under-21 international originally linked up with the Reds on a short-term deal in March and announced himself with goals in each of his first two games.
Edwards returns to the Checkatrade.com Stadium on a two-year contract, much to the delight of Crawley boss John Gregory.
"Other clubs were keen because he did well for us at the end of last season, but he likes this club," Gregory told the official website.
"We all know what Gwion can give us, but I think there is a lot more to come from him. He's at a great age to develop his game and really move his career forward, and I'm sure our supporters will be as pleased as I am that we have him now for the next two years."
Edwards, who leaves Swansea having failed to make a first-team appearance, believes his new club can challenge at the right end of the table in 2014/15.
"Obviously having been here before I know the set-up and the way the manager wants us to play," he said.
"I really enjoyed my time last season and to score my first League goal during that loan spell was a highlight of my career so far. Hopefully there are a few more highlights to come. I think there is a lot of potential here and I'm looking forward to helping Crawley kick on in League One this season."