Andy King pulled one back for the visitors, but a Barnes penalty secured a home win.
"We started well today, not slowly like we did against Barnsley and we deserved this result," Garcia told the club's official website.
"I wanted a reaction from the players and they did really well. I felt we played one of our best games of the season and it was certainly the best first half that we have played here.
"Leicester are one of the best teams in the league and they have a good squad with a great manager and are doing very well, but we only gave them one real chance – which they scored from, and to be fair I believe we played a lot better than them."
Brighton are 10th in the league, while Leicester drop to third.