Wayne Rooney grabbed the only goal of the game on the stroke of half time to secure all three points for the Toffees and seal a dream return to the side he departed for Manchester United 13 years ago.
Speaking after the final whistle, however, Koeman criticised his side for looking "nervous" and insisted that they "need to improve a lot".
"The points are really welcome because you don't know what to expect on the first day," he told BBC Sport. "It wasn't the performance we like.
"The team looked nervous on the ball - that's the reason I changed the system at half time. We need to improve a lot."
On Rooney, he added: "Wayne showed he is still one of the best in his position. He was very comfortable on the ball, made good decisions and scored a great goal."
Koeman's side next face a midweek test against Hajduk Split in the first leg of their Europa League playoff, before travelling to Manchester City when league action resumes on August 21.