The Gunners moved fourth in the Barclays Premier League table, leap-frogging Spurs, as Giroud's first-half winner was enough to hand them bragging rights and keep the pressure up at the top.
The France striker told Sky Sports 1: "Everybody knows it's a big game for us and the fans are really excited about this time of the year as well.
"We want to keep it up after our three wins and we did well this afternoon. We never gave up and we showed a real togetherness."
Arsenal took the lead after 23 minutes when Tomas Rosicky found Theo Walcott on the right, his neat cross reached Giroud on the edge of the six-yard box and the striker flicked the ball expertly past Tottenham goalkeeper Hugo Lloris.
Giroud added: "I used to do that in training so Theo knows me very well. I feel I know my team-mates and they know me much better so we find each other without opening our eyes."
Arsenal manager Arsene Wenger has made only a small dip in the summer transfer market compared to big-spending Tottenham.
Wenger has completed only the free transfers of midfielder Mathieu Flamini and France Under-21 striker Yaya Sanogo.
On keeping his faith with the team, Aaron Ramsey said on Sky Sports 1: "Absolutely, he recognises the quality in the team and we've shown again today that we have an enormous anoint of quality in the team even though we're probably a bit short of numbers.
"That probably showed in the last 20 minutes of the game, we're a bit tired from all the games we have played.
"But he believes in us and hopefully we've repaid him so far and hopefully we can continue this good run we're on at the moment."
On the prospect of new faces arriving before the transfer window closes on Monday night, Ramsey added: "The manager's trying to bring in some players but he believes in the quality we have. We've shown again what we're capable of doing.
"I thought we played a tremendous game today, we worked ever so hard and the big man up front started it off for us and the whole team did very well today.
"We didn't give them much space to play through the lines and obviously we got the goal in the first half and defended ever so well and I thought we deserved the win in the end."
Wenger played down the fact that Jack Wilshere left the field minutes before half-time for Mathieu Flamini, who rejoined Arsenal this week on a free transfer from AC Milan.
The Arsenal boss confirmed that the England midfielder was suffering from an illness ahead of the game along with Bacary Sagna, who started from the bench in the closing minutes.
It could have been another worry for England boss Roy Hodgson ahead of the World Cup qualifiers against Moldova and Ukraine in the next 10 days.
Wenger said: "It was an illness, we had two sick players before the game.
"One became worse just before the game which was Sagna, and Wilshere was a bit better so I decided to play Wilshere and not Sagna, who couldn't play."
He admitted it was a tough test of spirit for his side today, saying: "I always praised our spirit and they proved me right today.
"At times especially in the second half technically we dropped a level.
"But overall, to sum it up their goalkeeper was their best player and we had plenty of chances but I concede that we did hang on in the end because we just wanted to keep the score.
"There were less goals than usual but still a very interesting derby."
On the Gunners winning bragging rights for their fans, Wenger added: "We know it's a special rivalry (for the fans), but we got the three points first, but it's a very special game for everybody here."
Wenger has a strong belief in his Gunners side and admitted his faith paid off today shown through the Arsenal players.
He said: "I believe the confidence factor showed through the players and when they have the right attitude for this club it's very, very important.
"Giroud in my opinion is a very strong centre-forward and he has shown that."
Arsenal are expected to be late movers ahead of Monday's transfer deadline - with Karim Benzema, Mesut Ozil and Angel di Maria of Real Madrid said to be top of the wanted list.
Wenger added: "You will see what we will announce to you. If we sign somebody I promise we will be very quick to announce it....maybe we will have a good surprise for you.
"I believe ideally we still want strength in the squad because of long-term injuries and you always want more quality, but we are working very hard and let's give us time until tomorrow at 11pm."