Chelsea peppered the home goal with 13 corners in the first half but looked set to be frustrated as the interval approached, only for Doucure to bring Chilwell down in the box and gift Thomas Tuchel's side a penalty.
It was redemption for Italy midfielder Jorginho, who saw his penalty saved by Everton and England goalkeeper Jordan Pickford during the Euro 2020 final last July.
A brilliant save from Edouard Mendy early in the second half to deny Abdoulaye Doucoure preserved Chelsea's advantage, and that was as good as it got for a disappointing Everton, who only survived relegation by the skin of their teeth last term.
England forward Raheem Sterling could have marked an impressive debut with a late second but saw his close-range effort deflected just wide.
It mattered little as Chelsea, following a close season of upheaval after the sale of the club to an investment group led by LA Dodgers part-owner Todd Boehly, saw out the three points with ease.
"A win is a win, that is what we play for," Chelsea manager Thomas Tuchel said. "It is important to build confidence and belief now. We are open for improvement and criticism, but there are good things to take away.
"We got three points, we got a clean sheet. From here we go. We did not take the risks we normally do in the last 25 metres."
Coach Frank Lampard called for better from his Everton team this term in his pre-match programme notes, but a woeful pass from Ben Godfrey saw him forced off injured having made a last-ditch recovery tackle after just 15 minutes.
Everton later confirmed that Godfrey had been taken to hospital as a result of a lower leg injury.