The striker, 19, only joined on loan from Chelsea on Thursday but vowed to deliver the goals that would help the Swans' push for the Premier League.
After the win, which took Brendan Rodgers' men to within a point of second-place Norwich, Italy Under-21 ace Borini said: "They were my first senior goals.
"I played eight times for Chelsea and didn't score. Today I had chances to score and I've taken them.
"But the most important thing was the points to keep us in the play-offs and pushing for the top two. It's like eight finals left but we are up there and we will fight for every game."