Anthony was hurt in a challenge involving Inigo Calderon which resulted in Brighton's Spanish defender receiving a straight red card.
Edinburgh went out of his way to stress there was no malice involved and refused to blame Calderon for the injury.
He said: "There was no malice and I'm really disappointed for Byron because this has taken the gloss off a really great win."
There was an aubible crack as the players collided and former Spurs star Edinburgh added: "It was not very nice to witness that and I had a good view.
"It is a clean break, not a dislocation and I've spoken to him on the phone in hospital where he is stable.
"This is without doubt the greatest result for me as a manager and without doubt we're dedicating it to Byron."
Edinburgh was happy to go along with the suggestion that this was Newport's finest cup win for 45 years.
"It was thoroughly well-deserved," he said. "We scored two great goals at crucial times."
Brighton's new head coach Oscar Garcia was left feeling frustrated to have lost his first home game since taking over from Gus Poyet.
The Spaniard said: "We had many chances to score and win the game but Newport were more accurate than us."
Garcia also insisted that there was no intent in the tackle that resulted in Anthony breaking his leg.
He added: "It was a 50-50 challenge for me. I've seen it on video and Calderon wants to touch the ball and not injure anybody.
"The red card completely changed the game, it's not easy to play with 10 players."