Albion, after losing their opening three league and cup games, followed up a 1-0 win at Birmingham in convincing style to leave Garcia full of praise for the supporters for sticking by the team.
The Spaniard said: "I am very happy to get my first win at home and to get it in front of our fans makes it even better.
"We wanted the win today and I know that the fans are happy. Looking back at the start we made, it was difficult because we couldn't practice many days."
Burnley were forced to play the final 33 minutes with 10 men after keeper Tom Heaton was sent off for handling the ball twice inside the area in the process of releasing it under pressure from Ulloa.