Palace had not even scored a Premier League goal or registered a point before the match.
But an own goal from Cesar Azpilicueta and Wilfried Zaha's winner claimed the Eagles three crucial points.
In further bad news for Conte, Chelsea's Victor Moses picked up an injury which led to him being replaced by Davide Zappacosta in the first-half.
"I don’t like to lose, especially twice in a row," Conte told the official Chelsea website.
"For sure today my soul is bad and it will be difficult for me to sleep.
"But we must have the same sensation, me and my players, to turn a negative situation into a positive."
Conte refused to dwell on the injuries which had robed him of key players before and during the match.
"For us it’s a very difficult moment because we started the game without [Alvaro] Morata and [N'Golo] Kante, which isn’t simple," he added.
"During the game another important player, Moses, had an injury, and I hope to have him back very soon.
"This moment is very difficult for us, to lose three or four players is a big problem but despite this we have to find the will to fight.
"This season will be very difficult so we have to give 150 per cent, not just 100 per cent, like last season, because that’s not enough.
"My job is to be focused on my work with my players and try to do our best.
"Now, for sure, it’s not an easy moment for us but I think it’s important to find a solution and not look for excuses."
Next up for the Blues is a Champions League tie against Roma at Stamford Bridge on Wednesday (19:45).
"I don’t know if Moses' injury is serious, I hope not," said Conte.
"Drinkwater is still out. I hope to have Morata back for the game."
