After an uneventful first-half, the striker broke the deadlock from the penalty spot shortly after the restart before netting his and United's second from close-range in the final stages of the contest.
Dominic Calvert-Lewin's stoppage-time strike set up a frantic finish to the Premier League clash but it would prove a consolation for the visitors as United held on for maximum points.
Steve Bruce made one change to the side which were held to a 1-1 draw at Wolverhampton Wanderers last week, with Sean Longstaff coming in for Ryan Fraser, the 23-year-old's first Premier League start this season.
Carlo Ancelotti's side, requiring a win to go top of the table, were without attacking duo James Rodríguez and Richarlison through injury and suspension respectively while former Sunderland goalkeeper Jordan Pickford was dropped to the bench, with Robin Olsen making his Premier League debut for the Toffees.
The first goalscoring opportunity fell to the hosts in the early stages of the top-flight clash as Wilson raced onto Fabian Schär's through ball, getting in behind the Everton defence but the striker could only direct a side-footed attempt wide of the target.
That would be the only shot at goal until Allan Saint-Maximin went close to breaking the deadlock towards the half-hour mark. After Miguel Almirón brilliantly managed to get away from André Gomes, which began inside his own box, the Paraguayan found Wilson who threaded a neat pass into Saint-Maximin's path but the Frenchman's stinging effort at goal was blocked by a diving Olsen.
Dominic Calvert-Lewin, who had netted seven goals in his opening six Premier League games before today's meeting, had his first real sight at goal in the final minutes of the opening half but the 23-year-old's fierce shot was bravely blocked before midfielder Fabian Delph then fired well wide of the target.
The game sprung to life in the 56th minute as Wilson gave Newcastle the lead from the penalty spot. As Longstaff's corner was sent towards the front post, Wilson was fouled inside the penalty area by Gomes, with referee Stuart Atwell pointing to the spot.
Despite a lengthy delay before the spot-kick due to a thorough VAR check, Wilson stepped up and converted with ease, sending Olsen the wrong way as he side-footed the ball home to register his fifth league goal this term and the 28-year-old's third from 12 yards after successful spot-kicks against Tottenham Hotspur and Burnley.
Newcastle almost doubled their lead in instant fashion as Longstaff went narrowly close to completing a neat move from the Magpies, turning inside the penalty area before firing towards goal, with Olsen somehow managing to turn the effort over the bar.
Gylfi Sigurðsson's floated free-kick was flicked towards goal by Calvert-Lewin but the striker's header dropped kindly into Darlow's grasp.
Newcastle defended valiantly in the closing stages and, as the Blues surged forward in their hope for an equaliser, the Magpies added a second goal in the 85th minute.
As substitute Ryan Fraser raced onto Jamal Lewis' lofted pass, the Scotland international held off the rushing Yerry Mina before picking out Wilson inside the six-yard area, who tapped the ball home from close-range in front of the Gallowgate End to register his sixth goal of the season.
In the first of six stoppage-time minutes, Newcastle's hopes of a first league clean sheet at home were dashed as Calvert-Lewin netted for the Toffees, poking the ball past Darlow after Alex Iwobi's teasing low cross from the right flank.
Everton almost repeated Newcastle's heroics from the previous fixture between the two sides, which saw Florian Lejeune score two injury-time goals during a dramatic 2-2 draw at Goodison Park in January, as Bernard went close to grabbing an equaliser in the dying moments.
The Brazilian substitute's cross from the left flank flew towards goal, forcing Darlow to back pedal befor pushing the ball over his own crossbar.
Newcastle United: Karl Darlow; Jacob Murphy, Fabian Schär, Jamaal Lascelles (c), Federico Fernández, Jamal Lewis; Sean Longstaff, Jeff Hendrick, Miguel Almirón (Isaac Hayden 83), Allan Saint-Maximin (Ryan Fraser 74), Callum Wilson (Andy Carroll 87).
Subs not used: Mark Gillespie, Javier Manquillo, Ciaran Clark, Joelinton.
