At Stamford Bridge, Nicolas Jackson, Cole Palmer, and Mykhailo Mudryk scored for the home team.
Chelsea is still in eleventh position, but they are just four points behind West Ham in seventh, so they should at least secure their place in the Europa Conference League.
After suffering their 12th loss of the year, Newcastle is now only one point ahead of Chelsea in 10th place.
As Newcastle, backed by Saudi Arabia, did not capitalise on their fourth-place position from the previous season to earn their place in the Champions League for the first time in 20 years, Eddie Howe's pressure is also mounting.
However, Pochettino was up against a more outrage from inside his own team.
During last weekend's 2-2 draw with Brentford, Chelsea supporters demanded that the Argentine be fired.
The former Tottenham manager's first season in charge has not gone well; last month, the team lost the League Cup final to a seriously injured Liverpool.
However, Pochettino was the reason for hope as some of his highly criticised young players appeared to be developing.
The season suffered Chelsea, though, as they were unable to build on their strong start, allowing Newcastle to dominate the remainder of the first half.
Anthony Gordon's forced substitution halfway through the first period dealt a blow to the visitors' injury report.
Two minutes before halftime, though, Alexander Isak equalised the score for Newcastle with a stunning finish that he dug low into the far corner.
Palmer has been the only new addition Chelsea has been able to depend on this season, and the England international decisively gave his team the lead again shortly before the hour point.
After less than six minutes, Jackson deftly flicked in Palmer's shot to bring his season total to 12 goals.
Palmer's 11th Premier League goal of the year came when he received a pass from Enzo Fernandez, cut inside, and scored with great force.
On the other hand, since arriving for £88 million ($113 million) a little over a year ago, Mudryk has repeatedly angered various Chelsea coaches.
After just five minutes of play, the Ukrainian striker scored his sixth goal in 46 Chelsea appearances by using his lightning-fast speed to break through the Newcastle defence.
The hosts had a tense finish thanks to a rocket of a shot from Jacob Murphy into the top corner.
But in the latest disappointment of a disappointing campaign, it was too little, too late for the visitors.
