Pickford gifted Palace an equalizer in the 51st minute by letting a low shot from Christian Benteke squeeze under his body as he went down for the save.
Crystal Palace scored a late goal as Manchester City were held to a 2-2 draw while Spurs and Arsenal drew 0-0 and 1-1 against Watford and Sheffield respectively.
The move to Crystal Palace could boost Cenk Tosun's chances of starting for Turkey in this year's European Championship.
Wilfried Zaha equalized in the 82nd minute for Palace with his first goal of the season after Sadio Mane's opener in the 49th.
Palace's defence survived a Chelsea onslaught at Stamford Bridge in the first half — the Blues recording 11 shots to zero for Palace at the break.
Turkish defender Caglar Söyüncü opened the scoring with his first goal in the league in the 57th minute, when he was left free to head in after a deflected corner.
Gabriel Jesus and David Silva scored within 95 seconds of each other in the first half to help Manchester City beat Crystal Palace 2-0.
Five days after Paul Pogba had a penalty saved in a 1-1 draw at Wolverhampton Wanderers, Marcus Rashford regained spot-kick duties but struck his attempt against the post in the 70th minute.
United was without its entire first-choice front three as well as key midfielders Nemanja Matic and Ander Herrera.
Connor Wickham and Andros Townsend scored for Crystal Palace in their 2-0 win over Tottenham Hotspur in the fourth round of the FA Cup.
While Vardy's lone goal saw Leicester hand Chelsea their first home defeat this season, Hodgson's Palace stunned Pep's City 3-2 to end their 100 per cent home record.
Chelsea joined Liverpool and Watford in having a 100 percent record after three games, having previously beaten Huddersfield and Arsenal.
As West Ham scraps to stay in the Premier League, sitting three points above the relegation zone, resentment has grown over the decision to uproot from the Upton Park.
The win lifted United back above Liverpool in the race for Champions League qualification - five days before the fierce rivals meet at Old Trafford.
Raheem Sterling gave City lead inside a minute - he has now scored the earliest and latest goals in the league this season - before Christian Kabasele's own goal and a close-range finish from Sergio Aguero put City 3-1 up.
Playing four days after beating Real Madrid in the Champions League, Tottenham was without forward Dele Alli, centre-back Toby Alderweireld and goalkeeper Hugo Lloris because of injury.
Chelsea have endured a poor start to the season and are currently fifth in the EPL table with 13 points from eight matches. The defeat away to Crystal Palace was their third in the EPL this season.
Lukaku was kept relatively quiet at Old Trafford until the 86th minute when he tapped his 11th goal in 10 games for his new club.
Manchester City has 16 goals in its last three league matches and has a club-record 21 for the season after six games.
Leicester couldn't handle Coutinho's passing and movement in the first half, and only came into the game when he was off the field.
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