The Gunners closed to within four points of Liverpool at the Premier League summit.
While they have played a game more, Rice knows that points on the board are so important.
He told BBC Radio 5 Live: "We could have probably had 10 tonight.
"So many clear-cut chances which is a real positive, but to not score more is disappointing - to win 2-1 in a north London derby is a big thing.
"To close the gap is important - a big night for us."
He then added to TNT Sports about Myles Lewis-Skelly: "We play so many games now that we don't get much time to practice (set pieces) I just know now the area that I need to put the ball into and it is just about repetition.
"Spurs are a transition team. They play the ball short and continue passing. We win the ball high and they are wide open, and then Leo (Trossard) on his left or right foot is deadly.
"Tonight I was probably more of a box to box role. I just get excited and get that extra energy to run and run. With our centre halves you can be sure that I am allowed to attack because those two clean up everything.
"The young boy Myles (Lewis-Skelly) was unbelievable. For 18 years of age to be playing how he is... It is just ridiculous. The young players now have no fear, none whatsoever. Four or five times in the second half he used his body to get away from someone. He has that Moussa Dembele type strength. It is risky but to show that personality and character at 18 is unbelievable."