Media playback is not supported on this device
Spurs needed a point to finish second, and above Arsenal for the first time since 1995 - but failed as their north London rivals beat Aston Villa 4-0.
Pochettino said: "I think today we showed that this wasn't a team that had played well through the whole season.
"I think this was my worst day as a manager in England or Spain."
Spurs, who had 62.5% of the possession, were repeatedly caught out on the break by their opponents, who had already been relegated.
They were 2-0 down at half-time, but hopes of a recovery increased as first Erik Lamela pulled one back, and then Newcastle striker Aleksandar Mitrovic was sent off.
Instead, Tottenham conceded a penalty, converted by Georginio Wijnaldum, before two late goals saw Newcastle complete a stunning win. The result left Pochettino's side third in the Premier League.
When asked about what Newcastle manager Rafael Benitez said to him at the end of the game, Pochettino said: "It was strange - Newcastle are relegated but he felt sorry for me."
Arsenal's second place was their best placing in the Premier League since the 2004-05 season.
Media playback is not supported on this device
An Olivier Giroud hat-trick helped the north London side sweep aside relegated Villa for their 20th victory of the campaign.
On hearing the result at St James' Park, Gunners fans celebrated 'St Totteringham Day' on social media - the point in the season where Spurs can no longer finish above Arsenal.
Their manager Arsene Wenger played down the achievement, saying: "We don't start the season and think we want to finish above Tottenham but today we played with the only target we had. Today the aim before the game was to finish second.
"We achieved it. For a while our supporters were not going home very happy and today at least they had some satisfaction."
Mauricio Pochettino said he suffered the lowest point of his managerial career as his Tottenham side fell to a 5-1 Premier League defeat at Newcastle.