Thomas Vermaelen’s dramatic 95th-minute winner against Newcastle United on Monday night secured a fifth successive Premier League victory for the Gunners and ensured that the gap between Arsenal and third-placed Spurs was narrowed to one point.
And the England international says the club must build upon the victory and finish the season strongly to seal a Champions League place.
"We want to win and make this club into a club it should be," he told the club's official website.
"It's been an up-and-down season but it's the way you finish that matters. Tottenham, we're breathing down their necks now and hopefully they can feel it.
"I am sure they are slightly worried. They have some top-class players as well and no game is easy in the Premier League. We will find it tough going away to Everton, so we will see what happens."
The victory over Newcastle completed a remarkable turnaround in form for Arsenal, who were once 10 points behind their fierce rivals and threatened to fall a massive 13 points adrift when Spurs took a two-goal lead at Emirates Stadium.
It was also the first time a team had come from behind to win four successive Premier League games, and Walcott added that finishing above Tottenham would be hugely satisfying.
"It's really important [to claim third] and hopefully we finish above Spurs as well," he said. "That is an even bigger bonus for the fans especially. It will be a very good year definitely.
"We won't get too carried away with this result [against Newcastle], we just want to push on and keep getting the three points. Then we will see where it takes us."
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