The Gunners were made to hold on after conceding to a Harry Kane goal early in the second half, before Alexandre Lacazette squandered two opportunities from the bench to salvage a share of the spoils.
Lacazette sent the first opening over the bar via a volley and shot wide when played through in added time at Wembley Stadium, meaning bragging rights for Tottenham and a seven-point advantage over their city rivals.
Cech conceded after the game that Arsenal only have themselves to blame for coming away without a point, telling reporters: "In this type of game you need to take your chances, and I know we had two at the end. But unfortunately we did not take them.
"As I said before the game, it was a game which had a huge importance. Not only because it was a North London derby and obviously this is the most important game for Arsenal fans. But obviously for the position in the table we were behind, we needed to close the gap and we've failed to do so."
Cech has now gone nine Premier League games without a clean sheet - his longest-ever run in the competition.