The stubborn National League side held out for nearly an hour before a Salomon Rondon double saw the Premier League side progess.
It was a magnificent journey for Boreham Wood and they can be mightily proud of their performance at Goodison Park.
There was an emotional moment before kick-off as Everton's Ukraine star Vitalii Mykolenko led his side out as captain, with both teams draped in flags of his homeland.
A display of unity was made before the game as John Lennon's iconic hit Imagine echoed around Goodison Park.
Everton dominated and created a couple of chances though Mykolenko and Anthony Gordon but nothing to really trouble Boreham Wood goalkeeper Taye Ashby-Hammond.
The game was paused briefly three minutes before half-time due to a medical emergency in the Lower Bullens Road stand with Everton's medical staff dashing across the pitch to help out.
The National League side showed exactly why they had made it to the fifth round and went into the break 0-0.
"I thought they were excellent, especially defensively," Kevin Campbell told talkSPORT. "They are very disciplined in how they defend.
"Everton have had opportunities, but they haven't taken them."
With frustration growing around Goodison Park, Rondon - having wastefully headed over one Jonjoe Kenny cross - made amends in the 57th minute.
Kenny again supplied the delivery, this time along the ground, and the Everton striker nipped in front of his marker to beat Ashby-Hammond at his near post.
Richarlison thought he added a second to give Everton some breathing space, but VAR spotted a handball by the Brazilian and ruled it out.
There was a nice moment for diehard Toffees fan Kane Smith, who received a standing ovation from Goodison Park as he limped off injured.
Everton made sure of their progress when Ashby-Hammond could not prevent Rondon's 84th-minute header from crossing the line.
Frank Lampard's side will face Crystal Palace in the quarter-final but Boreham Wood can be mightily proud of their run to the fifth round.