Papiss Cisse had struck in the second minute for the Magpies, who ultimately looked toothless in attack, with Demba Ba on the verge of completing a £7 million move to Chelsea
The Magpies got off to the perfect start as Papiss Cisse looped a header over Tim Howard, but Leighton Baines hit back just before the break with a superbly-struck free kick from 30 yards.
Substitute Victor Anichebe snatched the points for the Toffees immediately after coming on in the second half, firing home with his first touch.
Demba Ba was absent from the hosts' side as he was in London to discuss terms with Chelsea over a prospective £7 million move and replaced by Shola Ameobi, while Marouane Fellaini's return from suspension was a boost for David Moyes.
The Magpies' top scorer was certainly not missed just two minutes in as Tim Krul's lofted free kick bounced over Ameobi into the path of Cisse, whose header drifted up and over Howard, but dropped just under the crossbar.
The Senegal No.9 just failed to get on the end of Ameobi's cross in a breathless opening six minutes, while Baines drew a great save out of Tim Krul with a well-hit free kick.
The Toffees grew in confidence as the half went on, Fellaini was only denied an equaliser by Davide Santon's brave block after slick work from Nikica Jelavic and Steven Pienaar on the left.
James Perch's flicked header from Vurnon Anita's free kick hit the post, but Everton continued to press and Baines' superb pass in the left channel set Pienaar through on Krul's goal but the Dutchman closed the angles well and smothered his effort.
The Toffees were finally level just three minutes before the break as Baines lined up a free kick from 30 yards and opted for power over placement, launching his effort in the far corner and wrong-footing Krul.
The second half started in a much cagier fashion, but the hosts created the first real chance as Ameobi got on the end of Marveaux's lofted pass, but could only prod the ball past the post.
The hosts responded perfectly and took the lead for first time moments later. Fellaini plucked a long ball out the air before feeding Jelavic out wide, the Croatian squared for substitute Anichebe who slotted home with his first meaningful touch.
The hosts were looking out of sorts and Mike Williamson almost scored an own-goal, but his skewed clearance was tipped over by Krul.
Cisse moved into a central position as Alan Pardew went all out for the win, Santon burst forward and could only hit Howard with a one-on-one effort, although team-mates were better placed centrally and Sylvain marveaux's half-volley from the resulting corner was blocked away.
Everton remained resolute in the closing stages, however, and held out for a win that sees them leapfrog Arsenal in the table into fifth.