County substitute Danny Crow cancelled out Ashley Barnes' opener before scoring again in the first period of extra time, while Conor Washington added a third to make sure of the result for the club who only secured promotion from the Conference National last season.
"We had many chances to score and win the game but Newport were more accurate than us," he said.
Newport defender Byron Anthony suffered a double fracture of his right leg at the Amex Stadium but Garcia did not think there was any malice in Inigo Calderon's challenge, despite the Spaniard being sent off as a result of the incident.
"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," he added.
"The red card completely changed the game. It's not easy to play with 10 players."
Newport manager Justin Edinburgh agreed with Garcia and dedicated the win to Anthony.
"There was no malice and I'm really disappointed for Byron because this has taken the gloss off a really great win," said the former Tottenham defender.
"It was not very nice to witness 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 we're dedicating it to Byron."