The England international made a sensational solo run before netting the opener, and his strike was followed up by Romelu Lukaku and Leon Osman in the second half.
Ross Barkley stunned St James's Park with a fantastic run halfway up the pitch before a decisive finish in the first half, further pressing his England claim.
The Toffees doubled their lead early in the second half when Romelu Lukaku pushed home from Gerard Deulofeu's cross before Leon Osman put the game to bed in the late stages.
The victory marks the first time that Everton has completed a league double over Newcastle since 1999 but, more importantly, moves it within six points of fourth-placed Arsenal - which it hosts April 6 - with a game in hand after the Londoners drew 2-2 with Swansea City.
|VIEW FROM ST JAMES'S PARK
|By Harry Slavin
Everton's assault on the Champions League places is still on track after a convincing win in the North East.
This fixture has historically produced a number of memorable goals, and Ross Barkley produced the pick of the bunch to give the Toffees a lead that they never looked like surrendering.
Romelu Lukaku continued his impressive record against Alan Pardew's side by notching his fourth in four games against them before Leon Osman capped off a the perfect performance for the Merseyside outfit.
All the good feeling after Papiss Cisse's late winner last time out evaporated as the hosts produced a disappointing display, and it is important that Newcastle rallies if it wants to avoid limping over the line this season.
But Everton quickly began to dictate matters and kept the hosts inside their own half for much of the opening stages.
The visitors' dominance looked set to pay dividends in the 16th minute as Osman played Lukaku through on goal, only for Newcastle goalkeeper Tim Krul to produce an outstanding save.
However, Everton's opener did soon arrive as Barkley lit up St James's Park with a contender for goal of the season. The 20-year-old picked up the ball from well inside his own half after a Magpies corner, sprinting all the way to the edge of the area before cutting inside past a string of recovering defenders and unleashing a thunderous left-footed strike into the top corner.
Newcastle looked to respond and demonstrated plenty of industry but their lack of creativity in the final third was their undoing as they struggled to test Howard.
The hosts' failure to break down the Everton defense was in stark contrast to the pace and guile of Martinez's men, who moved into a commanding position shortly after the interval.
On-loan Barcelona winger Gerard Deulofeu fired a warning shot by flying down the right and drawing a strong save from Krul.
Yet Newcastle did not learn from that close call and found itself two behind in the 52nd minute as Deulofeu latched onto a clever ball from Osman and delivered a precise low right-wing cross for Lukaku to tap home.
The hosts' disappointing evening was summed up 12 minutes from time, Vurnon Anita shooting wide with the goal gaping before Osman finished off another sweeping Everton attack to emphatically end their run of five away Premier League fixtures without a win.