Arsenal were themselves leading Newcastle 1-0 at the time but a late Newcastle equaliser would have condemned Arsenal to the Europa League.
"We weren't aware of the Spurs result and we knew what we had to do, it was in our hands and we did the job," he said.
"We had to prove people wrong. We had a disappointing result against Tottenham earlier this season and it's never nice to lose to them.
"A lot of people thought we wouldn't do it today [Sunday] so it's nice to show them that we could do it.
"I'd like to say it's a shame [that Spurs missed out on fourth place] but it's not really."
Asked about his aspirations for the club in the future, Walcott said: "Challenging for the title, that's our next step, our next goal.
"I'm very proud of everyone in that dressing room."