The result ended Newcastle's four-match winning streak and prevented Rafael Benitez's side from leapfrogging the Toffees in the standings.
"When you think of Everton he's the first man you think about".
"The controversy was caused by an error by one of our employees, which he obviously realises wasn't the right thing to do, " said Allardyce, who was accompanied in his news conference by the game's robot mascot, which allowed teenager Jack McLinden to fulfil the role "remotely" despite multiple health problems. I don't dwell on it. "We completely dominated the game", Allardyce said on Sky Sports.
The visitors saw the better of the ball in the closing stages as they looked to snatch a point, and they almost succeeded in the last minute of injury time, but Jacob Murphy's cross was headed behind strongly by Seamus Coleman as the Magpies' four-match winning streak came to an end.
"Breaking down a very stubborn Newcastle defensive unit, with all players in their own half trying to suck you in and hit you on the break, was hard. Normally if you go away against these kind of players and these kind of teams, it is not easy to get anything".
With little other than mid-table positioning at stake, Walcott's goal only briefly livened up a contest that was largely mired in a sluggish tempo and minimal attacking risk, although Allardyce will be no less satisfied to end a three-game league run without a win.
Everton now have 45 points from 35 games with three games left to play, while Newcastle United occupy the 10th position with four games left to play. They have very good players and they can make the difference.
Some fans have criticised Everton's performances under Allardyce, but the ex-Newcastle boss felt his side played well against his former team.