Everton frustrated the visitors in the first half and got their reward when Shaun Wright-Phillips lunged in on Tim Cahill on the edge of the box.
James Beattie powered in the penalty for his first League goal this season. Frank Lampard's fine effort made it 1-1 shortly after the break and, although Chelsea had a Didier Drogba strike ruled out for offside, Everton held on.
The hosts therefore avoided setting a new club record of seven consecutive defeats and stopped Chelsea making it 10 Premiership wins out of 10 this season. Everton fans will also hope the battling performance kick-starts a season that had been so disappointing thus far.
Moyes responded to his side's scoring problems by partnering the fit-again Beattie and Duncan Ferguson up front for the first time. Chelsea counterpart Jose Mourinho made only one change from the 4-0 thrashing of Real Betis in midweek, restoring Petr Cech in goal, and his side were immediately into their stride.
Michael Essien blazed over the bar and Nigel Martyn had to get down quickly to turn away Lampard's fierce drive - all inside five minutes. However, Everton settled and began to get to grips with the Premiership leaders.
They might even have taken the lead on 15 minutes when David Weir was left unmarked from a Mikel Arteta corner but he could not keep his header away from Cech. Beattie added to Chelsea's unease when, in the space of five minutes, he cut John Terry's nose in an aerial challenge and caught Claude Makelele on the head with a stray boot. Chelsea were rattled and Wright-Phillips' unnecessary challenge on Cahill was typical of their anxiety.
Goodison Park erupted as Beattie's powerful penalty clipped the underside of the bar and bounced in but the home fans endured some worrying moments before half-time. Joseph Yobo did well to block Asier del Horno's shot, the injured Nuno Valente had to be replaced by Matteo Ferrari and Tim Cahill was left hobbling after accidentally sliding into team-mate Tony Hibbert.
But just as they did against Bolton in their last Premiership outing, Chelsea went in at half-time a goal behind. On that occasion, they destroyed Sam Allardyce's men with five second-half goals and it took Lampard just over four minutes to level against Everton.
The England midfielder was allowed to run into space just outside the box and he bent a wicked shot just out of reach of Martyn's dive. But although Chelsea enjoyed plenty of possession, Everton refused to capitulate as Bolton had done. They rode their luck when Drogba's strike was disallowed for a dubious offside decision against Eidur Gudjohnsen but so did Chelsea when substitute Marcus Bent's cross hit Terry's arm in the penalty area.
Substitute Arjen Robben arrived to inject some pace and trickery into the Chelsea attack and the winger watched a dipping effort fly just over the crossbar, with Martyn beaten. The visitors piled forward in the dying minutes but they could not break down Everton, who were rightly given a huge ovation by the home fans at the final whistle, BBC Sport reports.
Rescuers found the pilot of one of the two Su-34 fighters that had collided in midair in the Far East on January 18