Leicester threatened a late comeback when Andy King rose unchallenged to head home from an Anthony Knockaert free kick to make it 2-1 - but Barnes netted his second from the penalty spot 13 minutes from time to clinch the points for Garcia's side.
Brighton's head coach considered the first-half showing in the midweek defeat against bottom-of-the-table Barnsley as the worst of the season by his side.
However, he was delighted to report a notable improvement when the leaders came to town.
The Spaniard said: "I want to mention all the players, not just one. On Tuesday it was our worst first half, here maybe it's the best first half at home.
"In this league you have to fight for every ball if you want to win a game.
"The atmosphere was very noisy. We wanted to start well and we wanted to be brave and we did it."
Garcia has no wish to talk up the Seagulls play-off challenge at this stage and insisted: "My players don't hear about the play-offs from me - they only hear from me about the next game.
"It will be a mistake for them to think of the play-offs.The time for that is at the end of the season or a month before."
Brighton's victory was achieved with them losing goalkeeper Tomasz Kuszczak with a stomach muscle injury in the warm-up, before skipper Gordon Greer was forced off early on with a calf problem.
Garcia added: "The difference between this game and when we played Barnsley was that we started really well.
"I wanted to start like this,we wanted to show a reaction and we did it.
"We have had a lot of injuries this season and it's not easy, but I'm happy with how we're managing it."