Wanderers had taken a first-half lead after Seagulls midfielder David Lopez steered a Chris Eagles corner into his own net, but Brighton hit back with three goals in four blistering second-half minutes to triumph 3-1.
Jay Spearing headed a Kazenga LuaLua free-kick into his own goal to bring the home side level, before Inigo Calderon and Will Buckley both added further efforts as Brighton stretched their unbeaten league run to six matches.
Garcia said: "It was a very important win for us. It was important more than anything to get three points after the draws at Reading and QPR.
"I told the players at half-time that that we needed to be more clinical. We lacked a killer instinct in front of goal despite a lot of pressure in the first half.
"When you play at home there is more pressure, more expectation, so I am pleased for the players because we have many injury problems and it has made it difficult for us."