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The American recorded his first victory since taking charge in October to lift the Swans off the bottom of the table and on to nine points from 13 games.
He said: "I see the win as a turning point in so many ways.
We felt pure ecstasy today
"Sometimes you feel like you need something crazy and maybe this game can be that moment."
Bradley added: "It has been a difficult season and a lot of things have not bounced the right way.
"But we still show up every day with a great mentality and I couldn't be happier for that group."
Swansea had led 3-1 but trailed 4-3 before Fernando Llorente scored twice in stoppage time to earn their second win of the season.
"I have been on both sides of games like that, so I understand how Palace feel," added the former USA coach.
"As far as we are concerned, it was a huge victory and one which the group thoroughly deserves.
"However, when I see the deflection off Jack Cork's head [for Palace's third goal], there is a moment when you think we just need some luck to turn things around.
"In the end the character of the players shone through and I am very happy for them."
Dutch midfielder Leroy Fer, who scored twice in two minutes to give Swansea a 3-1 lead, said: "This result is a good springboard, we have to keep going and moving forward.
"It was heartache for us when Everton equalised late on last week, but we felt pure ecstasy today."
Swansea's dramatic 5-4 win against Crystal Palace can revitalise the Welsh side's Premier League campaign, says manager Bob Bradley.