Callum Wilson scored his seventh and eighth goals in six matches against the Toffees, with Joelinton and substitute Jacob Murphy adding the others as the Magpies made it 10 goals in their last two matches.
Victory opened up an eight-point gap to fifth place but Howe, whose side thrashed Tottenham 6-1 on Sunday, is refusing to get ahead of himself.
“We know nothing is taken for granted from our perspective,” said Howe.
“It puts us in a lot stronger position. To get six points from Tottenham and Everton is a great return. It was always going to be a difficult game for us but we handled the occasion well in a hostile environment.
“The first goal was going to be crucial. It was probably our best move and it came at a good time for us. The confidence was there and it was evident in the second half, maybe a bit of the edge of the game had gone but I think we had to earn the right to get to that point because Everton put us under pressure in the first half.”
Everton manager Sean Dyche was unhappy his side not only failed to capitalise or build on an encouraging opening first quarter of the game but then capitulated too easily after the break as their run without a win extended to 10 matches.
“Very disappointed in the end. I thought the first half we did everything I wanted, other than concede, everything I thought was right about the performance against a good side in Newcastle,” said the Toffees boss, who saw Dwight McNeil briefly make it 3-1 late on.
“Mixing our play, the energy, the pressing, I thought it was all there.
“As soon as the second goes, which is really a poor goal defensively, it is how quickly the game gets away. We can’t let that happen.
“It happened against Fulham (the last home game which they lost 3-1), the whole performance changed on one goal and it happened again tonight so we have to correct that very quickly.”