Abdoulaye Doucoure cancelled out an effort from Joao Felix to leave the two teams level in the 69th minute, but Kai Havertz's late penalty looked to be securing all three points for Graham Potter's side, who were chasing a fourth straight win in all competitions.
Simms had other ideas, though, as the forward came off the bench to net his first goal for Everton, with the Toffees twice fighting back to secure a share of the spoils.
The result has kept Chelsea 10th in the Premier League table, 11 points behind fourth-placed Tottenham Hotspur, while Everton remain 15th, two points clear of the relegation zone.
The home side came close to taking the lead in the fifth minute when the ball fell for Mateo Kovacic inside the Everton box, but the midfielder's powerful effort was just wide of the post.
Havertz was next to try his luck for the home side, heading a corner over the crossbar in the 10th minute, with Potter's side making a strong start to the contest at Stamford Bridge.
Pickford then made a smart save to keep out a low strike from Felix in the 34th minute, with the home side looking the more likely to make the breakthrough at Stamford Bridge.
Chelsea had the opportunity to threaten from a free kick in the closing stages of the first period, with Dwight McNeil lunging to bring Enzo Fernandez to the ground, but the Argentine's effort was blocked, with Everton's strong defensive work allowing them to enter the half-time break all square.
Pickford made another stop to keep out a point-blank Havertz header early in the second period, with the German rising to meet a brilliant cross from Chilwell.
Chelsea finally made the breakthrough in the 52nd minute, though, and it was a first home goal for Felix, who managed to squeeze one into the far corner after Michael Keane had cleared a Chilwell cross to the on-loan Atletico Madrid attacker.
Christian Pulisic thought that he had doubled the home team's advantage in the 56th minute with a brilliant finish, but the offside flag came to Everton's rescue.
The Toffees had a chance to level just before the hour when Demarai Gray delivered a brilliant free kick into the home side's box, but Keane just could not make contact on the ball, before Kovacic had one deflected over for Chelsea during a lively period of the contest in London.
Everton were level in the 69th minute, though, and it was no surprise that it came from a corner, with Doucoure managing to angle a header into the back of the net after James Tarkowski had make first contact on a corner from McNeil.
Pickford kept out a strike from Havertz in the 72nd minute, with Chelsea looking to hit back, and the Blues were awarded a penalty in the 74th minute when Ben Godfrey brought Reece James to the ground with a clumsy challenge.
Havertz was the player to step forward for Chelsea, and the German sent Pickford the wrong way to make it 2-1 to Potter's team with little under 15 minutes of normal time remaining.
Everton managed to level the scores in the 89th minute, though, when Simms wrestled Kalidou Koulibaly off the ball before squeezing his effort under Kepa Arrizabalaga to score his first goal for the club.
Chelsea put the pressure on in the closing stages, with Everton having to survive some dangerous moments from corners, but the Toffees held firm to secure a welcome point in their battle against the drop.
Next up for Chelsea in the Premier League is a home game against Aston Villa on April 1, while Everton will welcome Tottenham Hotspur to Goodison Park on April 3.No Data Analysis info