Everton goalkeeper Jordan Pickford produced several incredible saves to deny Chelsea a second-half equaliser after Richarlison put the home team ahead in the 46th minute. Richarlison celebrated in front of the Everton fans following his crucial winner, throwing a flare back into the crowd.
"It is what I am there for and it is a team effort at the end of the day, it is all about getting three points," Pickford said.
"I really enjoyed it but that is what I am there to do, help the team. We cannot get too carried away, we have to keep fighting and the fans have played a massive part.
"The manager spoke at half-time and said to Richarlison to keep doing that and your chance will come. It shows working hard as a team and individuals and you'll get rewards."
Chelsea dominated possession for much of the game, but could not find their way through Everton's defence, who put in a resolute display in Lampard's first game against his former club since he left Stamford Bridge last year.
Everton almost got off to a dream start when Anthony Gordon charged forward from kick-off, winning a free-kick just outside Chelsea's box within 20 seconds but the resulting effort was curled over Edouard Mendy's post.
A scrappy half ensued, with no clear goal scoring chances for either team. Chelsea dominated possession but Everton challenged them on the counter, while also providing no way through at the back by defending with every man behind the ball.
The home side's efforts were rewarded immediately after the break, Richarlison calmly slotting the ball into the Chelsea net after pressuring Cesar Azpilicueta off the ball on the edge of the Chelsea box.
Chelsea's best chance came on 59 minutes when Mason Mount hammered a volley that ricocheted off both posts. Azpilicueta picked up the rebound and fired it towards Pickford's goal, but the England goalkeeper did brilliantly to run to the other side of his net to deny the visitors an equaliser, before producing another terrific stop to block Antonio Rudiger from close range.
Everton continued to push for a second, with chances falling to Gordon and Demarai Gray, before seven minutes were added on to set up a nervy period of extra time.
Pickford was called into action once again on 93 minutes to deny a close-range effort from Mateo Kovacic and Everton held on to secure the three points.