"It was a tremendous game and I think both managers will have enjoyed it hugely. David Moyes will probably be the most disappointed. From the first minute to the 90th minute it was end-to-end. It was a magnificently meaty game."
Marouane Fellaini marked his return from a one-match ban by scoring his seventh goal of the season with a fine finish before half-time but both sides had to settle for a point that did not satisfy either of their purposes.
Everton's bright start to the season has now stalled with six draws in their last eight Premier League games while Arsenal picked up their second point on the road in four days following
Saturday's goalless draw at Aston Villa.
Walcott was on target inside a minute when he exchanged passes with Aaron Ramsey and curled a finish high past Everton keeper Tim Howard with the help of a deflection.
David Moyes feels Everton 'deserved victory' against Arsenal
Arsenal's bright start suffered an early blow when Laurent Koscielny pulled up with what looked like a muscle injury and was replaced by Kieran Gibbs.
Everton were rocked back but were able to build momentum as the half progressed, with Fellaini inevitably the inspiration behind most of their best work.
The giant Belgian had a downward header saved by keeper Wojciech Szczesny before Everton pressurised Arsenal into a mistake and were rewarded with the equaliser after 28 minutes.
Steven Pienaar, another lively first-half figure, forced Bacary Sagna into conceding possession, allowing Fellaini to pick the ball up 20 yards out and sweep a low left-foot shot past Szczesny.
Theo Walcott's early opener against Everton was the 100th goal scored inside the first minute in Premier League history.
Everton ended the half on top as Nikica Jelavic turned superbly inside the area only to shoot over. The Croatian then brought an end to another dangerous attack when he unwisely interrupted a surging run from Leighton Baines by taking possession in what was clearly an offside position.
The second half produced more sparring than moments of genuine quality, although Szczesny reacted superbly to turn away Sylvain Distin's header from a Baines cross and Howard needed to react to block a snap shot from Santi Cazorla.
Everton draw 'was fair' - Arsene Wenger
Everton also had very presentable penalty claims ignored by referee Michael Oliver when Mikel Arteta appeared to drag Pienaar back as he advanced into the area, much to Goodison Park's obvious disgust.
Jelavic had suffered a frustrating evening but did provide two superb crosses only to see no team-mate on the end of his inviting deliveries.
Substitute Gervinho failed to get a touch on the end of Olivier Giroud's cross - but there was to be no winner of a largely undistinguished affair.
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