Garcia had failed to win any of his three games in charge, but Crofts' first goal since March 2010 did the trick.
The midfielder ghosted into the box to get on the end of a Kemy Agustien pass and bury memories of defeats to Leeds, Newport County and Derby.
Brighton did have to ride their luck after that, though, and Garcia was not in the mood to get too carried away.
"I am a pleased with the win but I am a person who does not look just at the result. I want to watch the overall performance," said the former Barcelona midfielder.
"We knew that our performances in the other games have been good but we have not been accurate in front of goal.
"At this level the most important thing is the points. We want to build a successful team but it will take time."
Garcia replaced the popular Gus Poyet at Brighton and talk of feeling the heat had already started in the wake of his poor start.
The former Maccabi Tel Aviv manager has a sense of perspective, though.
"After war in Tel Aviv this pressure is nothing!" he said.
"I want to dedicate the win to all the fans of Brighton, especially those who travelled to the game."
The 40-year-old also reserved a word for Republic of Ireland international Stephen Ward, who shone having signed from Wolves.
"Ward is an experienced player and comfortable on the ball," Garcia added.
"He is one of the players I am pleased to have in my squad."
It was a second home defeat of the season already for Birmingham, and manager Lee Clark was far from happy.
"It is frustrating and I cannot keep saying we deserved better from the game," he said.
"If we keep giving away disappointing goals like we did against Brighton it has to be frustrating.
"The players have already seen the goal on the footage because I was so angry and disappointed because it was so poor.
"We needed to talk about it as soon as possible.
"They had five players in the move who were never pressed and the deep midfield runner had a free run to score. You cannot do that at this level because you get punished.
"I am gutted for the fans because the support they have given the squad is unbelievable and all I want is to give them some positive results."