The Italian struck an 82nd-minute penalty to breathe new life into the Black Cats' relegation fight and end the Blues' 77-game unbeaten run at Stamford Bridge under Jose Mourinho.
Gus Poyet's men remain rooted to the foot of the Premier League table but, being one point adrift of safety with a game in hand over their rivals, Borini says that survival is very much on the cards.
He told BBC Sport: "That's why you take penalties, because you are confident and you see a goal.
"It was three points for the game and obviously Chelsea were up for the league and we showed that we have great character and great belief."
The North-East outfit conclude their Premier League campaign with home fixtures against West Bromwich Albion, Cardiff and Swansea City, either side of travelling to Manchester United on the penultimate gameweek of the season.
The West Londoners had looked to be cruising when Samuel Eto'o fired them into a 12th-minute lead, but Connor Wickham leveled matters for the visitors shortly after.