Dominic Calvert-Lewin netted a brace and Oumar Niasse also got on the scoresheet as the Toffees overcame the Championship side, to book their place in the fourth round of the Carabao Cup, where they will now face Chelsea.
Despite a bumper spending spree in the summer, it has been a tough start to the season for Koeman's men, who are languishing at the wrong end of the Premier League table.
However, four games on the bounce at Goodison Park presents the perfect opportunity to go on a winning run, and it started perfectly against the Black Cats last night.
"Overall positive," said Koeman. "If we can rest some players for the weekend even against a Championship club we know every game in every competition can be a difficult one. We worked hard, the boys did well and we scored three goals. It was a good game. Maybe we had chances to score more goals.
"Everybody knows we're in a difficult moment and scoring goals and winning games is the best medicine to get our confidence back. Let's hope it is a boost for the coming Saturday."