After a goalless 45 minutes, Wilson's impressive header - his 50th Premier League goal - sent the Magpies into the lead before the striker, who netted a brace in the reverse fixture against the Toffees at St. James' Park, confirmed maximum points with a clinical stoppage-time finish, his tenth league goal of the season.
Steve Bruce's side headed into the league clash without a league victory in nine games and faced an Everton side who had recorded five wins in their last seven outings in the English top-flight.
The first goalscoring opportunity fell to the Toffees as James Rodríguez's teasing cross from the left wing was headed towards goal by Yerry Mina but the centre-back's effort was steered wide of the target.
Richarlison then forced Darlow into an important stop after the Brazilian stepped inside from the left flank before his drilled strike was beaten away by the alert United goalkeeper.
Jonjo Shelvey, making his 100th Premier League appearance for the Magpies, went close to sending the visitors into the lead as the midfielder picked up the ball on the edge of the penalty area after sharp build-up play between Wilson and Miguel Almirón, before his low strike was deflected past the far post by Mina.
United struck the crossbar on the half-hour mark as Wilson saw a close-range header from Ryan Fraser's corner tipped onto the woodwork by former Sunderland goalkeeper Jordan Pickford.
Darlow was called into action shortly after to deny Rodríguez as Michael Keane's lofted ball was controlled by Dominic Calvert-Lewin before the striker produced a neat back heeled pass into the Colombian's path but the United shot-stopper stood tall to block the dangerous effort.
In the early stages of the second half, Wilson went narrowly close to opening the scoring for the Magpies as Fraser's accurate cross fell into the path of the striker but could only place his low first-time effort wide of the target.
Darlow produced a key save to prevent Calvert-Lewin, who had previously netted five goals in his last four games against Newcastle, as the striker's curling effort was tipped away from danger by the Newcastle goalkeeper.
In the 73rd minute, Wilson broke the deadlock for the Magpies with his ninth goal of the season. After a teasing corner from Shelvey, Wilson rose high and brilliantly directed a powerful header into the far corner.
The forward, a threat to the Toffees defence all afternoon, almost doubled Newcastle's lead in the closing stages after racing onto Almirón's neat through ball but, after rounding Pickford, saw his effort from a tight angle strike the post.
However, Wilson sealed the Magpies' first victory at Goodison Park since September 2010 in the third minute of injury-time after a fine counter-attacking move.
After holding off a late attacking surge from the Toffees, Allan Saint-Maximin sent a dangerous through ball into the path of an overlapping Jamal Lewis before the left-back picked out Wilson who, free in the middle, buried a low effort beyond Pickford to secure Newcastle's first double over Everton since the 2001/02 campaign.
Newcastle United: Karl Darlow, Javier Manquillo, Isaac Hayden, Fabian Schär, Jamaal Lascelles (c) (Ciaran Clark 54), Jamal Lewis, Jonjo Shelvey, Jeff Hendrick, Miguel Almirón, Ryan Fraser (Allan Saint-Maximin 67), Callum Wilson.
Subs not used: Martin Dúbravka, Emil Krafth, Matty Longstaff, Elliot Anderson, Joelinton, Dwight Gayle, Andy Carroll.
