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.
"But during the last two weeks we haven't had a game mid-week and we've been able to practice more."
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.
Garcia stopped short of saying that this was a deserved dismissal, and said: "It was difficult to see."
Heaton received his first yellow card late in the first half for bringing down the on-rushing Will Buckley and Garcia admitted: "It was a quick decision by the referee. Maybe he saw another player near to Buckley.
"Some people have different opinions, but the only one that matters is the referee's."
Burnley boss Sean Dyche claimed that referee Darren Sheldrake had "factually" made the wrong decision in sending off Heaton as Buckley had touched the ball in between Heaton touching it twice himself.
Dyche said: "Factually I think he (Sheldrake) has made the wrong decision.
"I asked the referee for his view and he said it was for unsporting behaviour, throwing the ball up in the air and catching it twice. I asked him whether he had seen our player impeded in the area.
"The referee has got one chance to look at it and he made a quick decision.
"Maybe had he spoken to the linesman there may have been a different outcome.
"The referee also said to me that, in making the decision to send Heaton off, he wasn't affected by his first half decision not to."
This was Burnley's first defeat in five league and cup games this season and Dyche felt the sending off "changed the second half."
"In the first half we had the upper hand and we started the second nicely on the front foot."
Dyche confessed that, in terms of controversial sending offs that he has witnessed, Heaton's dismissal was "up there in the top five."
He added wryly: "You live and learn."