The England forward scored twice in February and netted his side's fifth in the latest win, which owed plenty to Emmanuel Adebayor's red card when Spurs were leading 1-0.
The result lifted the mood around the Emirates following Arsenal's worst start to a Premier League season under Arsene Wenger and took them above their north London rivals.
Walcott said: "We showed people how good we are. It'll give us tremendous spirit, we've got a lot of players back to fitness and everybody's very confident.
"If we show that courage and ambition every game we will climb up the table."
Walcott's goal was his third in the league this season on just his third start, and he said: "I've just wanted to play. Hopefully I'm showing people what I can do now."
Olivier Giroud was also on target with his fourth goal in as many games, and Walcott said: "People were saying things about Olivier at the start but he just needed time."
Giroud said: "It was an amazing atmosphere and I'm very happy to win my first derby.
"It is something special for our fans so I am proud. It was important for the future to win this game."