Against the run of play, Spurs took the lead through Rafael van der Vaart just before half-time, but Aaron Ramsey's tap in just after the restart made it 1-1.
The game seemed destined for a draw until the ball fell to Walker 17 minutes from time and the right-back hammered a bullet of a shot that flew past Wojciech Szczesny.
It was a strike that will live long in the memory of Spurs fans and of the 21-year-old himself as it was his first for the club.
Arsenal manager Arsene Wenger will feel hard done by given his team controlled much of the match, but Tottenham's superior finishing meant they prevailed to heap further pressure on the Frenchman, whose team have now suffered four league defeats from seven matches.
By contrast, Redknapp's team have won four straight games and are well and truly back on track for their Champions League assault after a wobbly start.
Just two days after celebrating his 15th anniversary in charge, Wenger, on the other hand, must now attempt to pick his team's spirits up and convince the doubters that his team are not in free fall.
Sunday's game was more than a battle for supremacy between the two north London giants.
The contest had many subplots - chief among them Emmanuel Adebayor's first game for Spurs against his former employers.
Gunners fans have never hid their dislike of the Togo marksman after his controversial celebration against them two years ago, and they greeted his every touch with huge boos early on.
The man himself looked pumped up for the contest. Tightly marked by Song, he galloped down the right flank early on but his cross was cut out before anyone could tap in.
Spurs should have gone ahead in the seventh minute when Song's misplaced pass allowed van der Vaart to play Scott Parker in but he drove straight at Szczesny.
Arsenal then went on to dominate for the next half hour but failed to make the pressure count.
Comfortable in possession, it looked only a matter of time before they scored.
But van Persie could only flick a header wide, Walcott's deflected shot crept past the post and Ramsey's header was hooked off the line by Benoit Assou-Ekotto and Spurs survived the onslaught.
The home fans became anxious at their team's failure to get a grip of the game and the frustration among the players came to the fore just before the half hour when Van der Vaart was booked for a late tackle on Kieran Gibbs.
The Gunners' best chance of the first half came soon after when Gervinho, in yards of space inside the box, side-footed wide from van Persie's cross.
The chance appeared to wake Tottenham. The hosts broke the deadlock in the 40th minute when van der Vaart chested down Adebayor's cross-field ball and volleyed past Szczesny.
The Dutchman ran across to celebrate in front of the jubilant Spurs fans while Wenger remonstrated with the fourth official, claiming van der Vaart had used his arm to control the ball.
Buoyed by the opener, Bale tried two audacious attempts that flew just over.
Their joy was to be shortlived, however, as Song, given too much space by van der Vaart, ventured in to the box from the byline and cut back for Ramsey who turned in.
Adebayor was handed a golden chance for an instant reply in the 57th minute when played clean through by Bale, but Szczesny saved brilliantly.
Bacary Sagna had to be replaced moments later after coming off badly in an aerial tussle with Benoit Assou-Ekotto which saw the Frenchman need to carried off on a stretcher.
The game went in to a bit of a lull until Walker sprung it in to life.
The ball fell to the youngster 30 yards out following Song's block on Luka Modric's shot and the full-back hammered a bullet of a shot which flew past Szczesny's out-stretched hand to make it 2-1.
Bale had a chance to kill the game moments later but he shinned an effort just wide when through on goal.
Arsenal's push for an equaliser left them exposed at the back but Szczesny was equal to the challenge, pulling off a world-class save to deny Defoe from close range.
The game ended on a sour note as Wenger and Clive Allen argued after the final whistle, with the Spurs coach claiming the Frenchman refused to shake hands with him before going down the tunnel.