The Blues looked to have turned a corner in Pochettino's first season in charge with a 4-1 win at Tottenham and a memorable 4-4 draw against champions Manchester City before the international break.
But the visitors crumbled in the second half at St. James' Park as Newcastle overcame an injury crisis that had left Eddie Howe without 13 first-team players.
"We had to come here, Chelsea, to show that it's going to be difficult for them to play, to win the game and to beat us," said Pochettino, who watched the game from the stands due to a touchline ban.
"But it was really easy in the way that we conceded and the way that we were so soft in every single challenge. We didn't show that we were playing for something important.
"That's what makes me angry and disappointed. We talk about that we are a young team and we have to learn, but I think these type of games make me very, very, very, very, very angry because it's about showing your personality and character."
Defeat leaves Chelsea down in 10th in the table despite spending over £1 billion ($1.3 billion) on new players in less than two years under the ownership of Todd Boehly's consortium.
Raheem Sterling had cancelled out Alexander Isak's early opener to leave the score level at half-time.
But Chelsea collapsed after the break as two goals in two minutes from Jamaal Lascelles and Joelinton put Newcastle in command.
Chelsea captain Reece James was then sent-off for two bookable offences before Antony Gordon rubbed salt in Pochettino's wounds late on.
"We thought that we were ready to compete today, but we didn't in the way that the competition demands," added Pochettino.
"Even if Newcastle weren't great, it was an easy win."
Newcastle's attention now turns to putting together a team strong enough to stave off elimination from the Champions League when they visit Paris Saint-Germain on Tuesday.
Defeat in the French capital will see the Magpies first experience of Europe's elite competition for 20 years end at the group stage.
"You look at the players who were missing and that was a giant performance from the players we have fit," said Howe.
"To be able to come together and give a performance like that speaks volumes for the character of the players we have, the leaders we have in the group and our ability to just focus on the present, on what's happening right now."
Newcastle climb up to sixth, three points off the top four.
