A Jorginho penalty deep into added time at the end of the first half was the only difference between the sides as Chelsea struggled to convert despite having 70% possession for most of the game.
"A win is a win and this is super important. We played at Everton and we've struggled over recent years to get points here," Chelsea boss Thomas Tuchel said after the game.
"We have things to improve but this isn't a surprise. Most of the time I'm happy with the first half, the second we lost too many balls and weren't as dominant. We struggled physically but showed the spirit to get the clean sheet."
"It [Godfrey injury] feels like it is a small fracture of his leg. We are assessing that. He will be out for a while, [Yerry] Mina is an ankle injury and could be out for a while," Lampard said after the game.
"When it rains, it pours."
As a result there were eight minutes added time at the end of the first 45 minutes and Chelsea capitalised when Abdoulaye Doucoure pushed Ben Chilwell over in the box. Jorginho stepped up and sent Pickford the wrong way to convert the penalty.
Raheem Sterling, who was making his Chelsea debut, did have the ball in the net a few minutes beforehand but it was ruled out for offside.
The 27-year-old had an excellent opportunity to tie up Chelsea's win on 82 minutes but Vitalii Mykolenko managed to get enough of his body in front of the ball to see it drift just wide of Pickford's net.
"Points are what it's all about. Early on in the season you want your performances to be up there -- that was the case today. We were dynamic and caused them problems," Everton defender James Tarkowski said after the game.
"I was excited going into the game. I've seen what this team can do. There are exciting times ahead, it showed in the performance today."