The Brazil international's 46th-minute strike secured the crucial three points as Frank Lampard's side triumphed in a critical match against the manager's former team.
However, Richarlison's goal would have meant little without a brilliant recovery save inches from the line from Cesar Azpilicueta’s shot by Pickford, who had been beaten by Mason Mount’s strike that struck both posts but rebounded out.
Lampard’s side were, for once, both effective and efficient at both ends of the pitch as they displayed a real desire to win with their 67-year top-flight status in real danger.
"We did brilliantly. The team, stadium, the fans were man of the match today," said Lampard afterwards. "From the drive in to the support, which we need. Thanks to them.
"The fans are in the middle of this. I am pleased the players could see on the way in what it means to the fans. They know that and understand it. They got behind us, were nicely aggressive, this place has to be a hard place to come. Results have been good here at Goodison before I came, we have to maintain that.
"They [the players] have to show they care. I prodded them a bit before the game because the situation is clear. We need to perform with that level in all our remaining games.
"Jordan made great saves. they are Chelsea and will have chances but we had chances as well. Jordan is such a great lad, he is England's No 1 and that is the reason why."
Home form was always likely to be Everton's route to salvation and 10 points from the last four games at Goodison have proved essential in just keeping them afloat in 18th place.
The recent form of Richarlison has been welcome with four goals in his last five league matches.
But arguably more important was the performance of Pickford, who after facing no shots on target for almost an hour, then produced an impromptu cameo to keep Chelsea at bay and the lead intact.
Everton set up in a 5-4-1 formation, with Alex Iwobi right wing-back and Seamus Coleman tucking inside, but far from being defensive it provided them with the security to harry and harass a Chelsea side who struggled to find any fluency.
Yerry Mina’s return at centre-back, after being rested for last weekend’s Merseyside derby defeat, also brought some much-needed assurance to the back five.
The first half ended with more yellow cards (five) than shots on target (one) but all that changed just 59 seconds after the interval.
Richarlison charged down an Azpilicueta clearance on the edge of his own area and the loose ball kindly dropped to Demarai Gray, who teed up his teammate to sweep a shot past Edouard Mendy.
Richarlison celebrated by grabbing a smoke canister thrown from the Gwladys Street end and hurling it back into the crowd.
The lead should have really been doubled moments later but left wing-back Vitalii Mykolenko blazed wide with only Mendy to beat.
Pickford then stepped forward with his superb goalline stop and then, with barely any time to think at the next phase of play, taking a shot from Antonio Rudiger full in the face.
He followed with a one-handed save from a Ruben Loftus-Cheek drive but with the pressure growing and nerves jangling Everton, buoyed by the inspirational form of their goalkeeper, put everything on the line to keep a clean sheet.
Chelsea managerThomas Tuchel told BBC Sport: "It is disappointing because over the last four matches we don't have enough points for what we want and it is concerning. We have one clean sheet over the last four matches and it is far too easy to score against us.
"We can't stop doing individual mistakes. We struggle to have repetitive top performances. Today was an okay performance but okay is not enough."