Brighton 2 - 0 Burnley
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.
Howe set for Cherries return
The favourite to take over at AFC Bournemouth is former manager Eddie Howe.
He left the Cherries nearly two years ago to manage Championship Burnley. Reports suggest Howe is keen to re-join family on the south coast and that the clubs are discussing compensation.
He's likely to be accompanied by his backroom staff - including assistant manager Jason Tindall.
Howe took training at Burnley as normal on Wednesday and this morning, as the clubs discuss a financial package.
The 34 year old made more than 200 appearances for Bournemouth and managed them for two years - during which time he masterminded promotion to League One.
Since his departure though, the club has failed to replicate such success, under former managers Lee Bradbury and most recently Paul Groves.