The Gunners trailed to Emmanuel Adebayor's early goal at the Emirates Stadium but after the Arsenal old boy was sent off, they roared back to win 5-2.
Walcott, who rounded off the win late on, told Sky Sports 2: "We showed people how good we are today [Saturday].
"It'll give us tremendous spirit, and we've got a lot of players back to fitness and everybody's very confident."
The dismissal of Adebayor, for a reckless lunge at Santi Cazorla, was undoubtedly crucial but Walcott backed referee Howard Webb's decision.
"I didn't really see it but I think he did go in quite two-footed and the referee's made that decision," the England international added.
Walcott's goal was just reward for an enterprising display which also saw him provide the cross for defender Per Mertesacker's equaliser.
"When you've got the big man up front there he's going to cause havoc for anyone," he said.
"I've just wanted to play and hopefully I'm showing people what I can do now."
Lukas Podolski, Olivier Giroud and Cazorla were also on the scoresheet and Walcott praised the France striker, who is starting to find his feet after a slow start following his summer move from Montpellier.
"Olivier's come in and people were saying things at the start but he just needed time," Walcott added.
Giroud, who was playing against Spurs for the first time, 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."