Llorente was linked with the exit door last week after failing to be included in the 18-man squad for the clash against Everton.
The Swans picked up a 1-1 draw on that occasion but Llorente was the hero on Saturday as he scored two goals in stoppage time after coming off the bench.
He had scored just one goal in 10 matches before his brace but now the former Sevilla star claims he wants to stay in Wales.
"I'm very happy here and I think I can help this team," he told reporters.
"It's difficult for me because of the language. I'd like to speak better than I do now, and I need to continue to study. After my (rib) injury, I lost my performance. But I always kept working in training and the coach saw this.
"These two goals are very important for me and my confidence and I feel very good. He (Bradley) has seen me play in other teams and knows I can do a lot of things. And I know I have to keep working for my best performance."