Danny Welbeck came off the bench to glance home an injury-time winner after Theo Walcott had equalised following Jamie Vardy's first-half penalty at Emirates Stadium.
The game's turning point was a red card for Leicester's right-back Danny Simpson after 54 minutes, when his side led 1-0, and Cech admits his side were handed a tough afternoon.
Speaking to BBC Sport, he said: "We had to concentrate in the final stages because any counter-attack was becoming more and more dangerous as we searched for the winner.
"It was tricky. What I like about our win is that it is always tricky when opponents are down to 10 men.
"They have good organisation and are fighting for their lives. We managed to break them down and create chances.
READ MORE | Welbeck: Winner was a beautiful feeling
"We kept missing chances but kept composure and kept coming. We had so many half-chances where we could have scored more."
Arsenal's win leaves them just two points behind league leaders Leicester and Cech believes the victory is vital in their bid for the title.
"If you have a cushion of five or eight points, it is a big difference," he added.
"We overturned eight points to two - it is significant difference."