Antonio Conte's side, who had come into the game boasting of a 10-point lead over second-placed Tottenham, started the London derby off in perfect fashion when Cesc Fabregas tapped in the opener after just five minutes.
But Wilfried Zaha and Christian Benteke's goals in the space of just three minutes turned the game on its head before Wayne Hennessey made a string of fine saves and Sakho pulled out two last-ditch clearances to secure the win for Palace.
While the result cut Chelsea's lead over Spurs to seven points following the north Londoners' 2-0 victory at Burnley, the three points helped Palace move away from the relegation zone.
And Sakho admitted after the match that Sam Allardyce's side had the travelling fans to thank for helping to inspire them onto a famous win over Chelsea.
"We played a good game. It was really hard but I think we played with a big heart," the defender told BBC Sport.
"Our fans really shouted for us and they gave us a lot of encouragement. We played 12 against 11.
"It was important to come and win here. We spoke at half-time about needing the three points. We weren't here to go back with a draw.
"We play with big determination. I'm so happy for us. It's a good team with good players and we are starting to believe. If you fight always like it's your last day, you can reverse everything."