A dull first half bore all the hallmarks of an end-of-season mid-table encounter with goalscoring opportunities at a premium.
However, the deadlock was broken soon after the break when Yannick Bolasie’s cross deflected off DeAndre Yedlin to Walcott, who took a touch before driving the ball into the roof of the net.
The game briefly opened up after that and Newcastle substitute Dwight Gayle should have done better when he hooked a shot over from close range before Seamus Coleman made an important headed clearance in stoppage time.
Ultimately, a lack of quality in the final third proved Newcastle’s undoing as they spurned several other opportunities to create something and Everton recorded a first win in four games to move onto 45 points.
Newcastle remain 10th on 41 points and need just one more point to be mathematically safe from relegation.
Allardyce has taken Everton to mid-table safety from just above the relegation zone, but there has been growing discontent among some supporters at what they perceive to be a negative playing style.
There were calls for Allardyce to be sacked during the 1-1 draw at Swansea earlier this month, with some fans unhappy with their side’s display, and his critics were unlikely to be appeased by this performance.
Everton seemed uncertain how to break down a disciplined Newcastle side, while Cenk Tosun found himself far too isolated from his team-mates in attack.