Gus Poyet's men earned just their third Premier League victory of the season in Thursday's encounter at Goodison Park, with Ki's first-half penalty ending Everton's 2013 unbeaten home league record.
Ki's first Premier League goal also curtailed rock-bottom Sunderland's six-match winless league run and left them three points adrift of safety.
With league fixtures against fellow strugglers Cardiff City, Aston Villa and Fulham on the horizon, Ki is keen to further boost the club's survival hopes.
"No one expected us to win, but we did. We had the belief among each other," he told The Chronicle.
"We prepared for the game like it was the last game of our lives. We did not think about Cardiff, just Everton. Now we can look towards Cardiff.
"Everton were unbeaten in 23 at home so it was a big challenge for us.
"Now it is very important we build on that win because we play teams near us in the table in Cardiff, Aston Villa and Fulham. These are the games from which we should take points.
"We are halfway through the season now and every game is huge for us. Gaining a first win against a team like Everton will give us a lot of confidence.
"We were disappointed after last weekend's goalless draw against Norwich because we should be winning games like that and West Ham.
"Yet we have beaten teams like Man City, Everton and Newcastle, which means we are capable of winning games."
