Victory also moved Spurs 17 points ahead of their north London rivals in the table and ensured they would finish above them for the first time since 1995.
Dele Alli's close-range finish and a Harry Kane penalty sealed the win, which could have been more emphatic had Alli and Christian Eriksen converted chances in the first half.
The result leaves Tottenham four points behind Chelsea at the top, while Arsene Wenger's side remain sixth, five points behind Manchester United and six back on fourth-placed Manchester City.
They play United at home next weekend.
"I feel angry and disappointed because obviously this is not a game we wanted to lose and this is a very big disappointment for us," Cech told Arsenal's official club website.
"It hurts because it was a huge game for both teams
They needed to win to keep their title chances alive
We needed to win to keep our Champions League finishing position [hopes] alive.
"Obviously we are very disappointed because it's a north London derby, it means a lot to the fans and it meant a lot to the club, to us
We needed three points and in the end we have none."
Tottenham confirmed last week they would move to Wembley next season while their new stadium is completed meaning this was the final north London derby at White Hart Lane.
It made victory all the sweeter for Spurs fans, who have waited 22 years to end a league season above their neighbours.
Cech insists finishing below Tottenham, however, is not chief among the squad's disappointments.
"Our target is to play for titles, so we are disappointed with our season because we aren't playing for a (league) title," Cech said.
"We have five games to play and we are out of the title race
It was not the objective (to finish below Tottenham), so we are disappointed with that.
"But Tottenham finishing above us or doing better is one thing, but I think the most important thing for the club, for the players, for everybody is to reach your own targets, not to worry about what the others are doing."
Chelsea's 3-0 win at Goodison Park earlier on Sunday may yet prove a decisive moment in the title race, given Everton were seen as their toughest opponents during the run-in.
Antonio Conte's side have Middlesbrough, West Brom, Watford and West Brom left to play, but Spurs could reduce the gap to one point given they visit West Ham first on Friday night.
They will then host Manchester United in what will be the last ever fixture at White Hart Lane.
"It was a very good game from everyone, from start to finish," Jan Vertonghen said of his side's latest victory.
"It was the perfect day to put on a performance and I'm very proud of it
We knew we had to focus
It was the last north London derby at the Lane and that's so special for so many people.
"I know what it means
It was a very big day and we've one more to come
Let's hope we can make that one special too."
Source : PA