United made a bright start to the game with Demi Stokes scoring her first goal for the club after arriving late at the back post, but that was eventually cancelled out by Emma Bissell's second-half strike.
Courtney Brosnan was the hero for Everton in the shoot-out, saving all three of United's penalties as the Toffees scored each of their spot-kicks.
Newcastle made sure to let Everton know they were in for a game right from kick-off with constant pressure and an eventual Stokes strike after just five minutes.
The left-back stole in unmarked at the back post before latching onto a perfectly-placed low cross from Sophie Haywood to an eruption of noise from the stands.
After relinquishing some possession following Stokes' goal, it was United who fashioned the next opening 18 minutes later.
A long ball from Charlotte Potts found Isabella Sibley but the winger couldn't lift the ball over Brosnan, who'd ventured off her line.
Newcastle continued to cause problems down the right through Sibley who took her time to pick out an inch-perfect cross to Katie Barker, but United's number nine could only blaze over from close range.
Just five minutes into the second half Everton found themselves level when Bissell diverted the ball beyond Hannah Hawkins after several deflections in the box.
United certainly didn't let that goal knock their confidence, though, with Barker finding space behind the Toffees' backline three minutes after conceding.
Newcastle's number nine appeared to have a clear strike on goal, but a last-ditch tackle saw her effort fall beyond the far post.
Both sides made several changes throughout the second half which made the game more stop-start. Elysia Boddy tried her look from distance in the final ten minutes but the ball was safely grabbed by Brosnan.
Just minutes into her first appearance of the season Georgia Gibson would've been wheeling away in celebration following an edge-of-the-box volley, but for a stunning Brosnan fingertip save over the bar.
Neither side looked to be settling for penalties with the Toffees breaking through United's backline in the final two minutes as Veatriki Sarri's low effort went narrowly wide.
The Toffees almost snatched all three points in the dying seconds of added time when Elise Stenevik's header crashed off the post and after a rebound, ended up in the arms of Hawkins.
Smart saves from Brosnan in the shoot-out denied Amy Andrews, Georgia Gibson and Rachel Furness to ensure the Toffees picked up two points from their journey up to the North East.
Newcastle United Women: Hannah Hawkins, Liv Watt, Charlotte Potts, Rachel Furness, Demi Stokes (Lois Joel 65), Amber-Keegan Stobbs (Elysia Boddy 56), Hannah Greenwood (Georgia Gibson 74), Isabella Sibley, Jasmine McQuade (Amy Andrews 65), Katie Barker, Sophie Haywood (Emma Kelly 74)
Subs not used: Grace Donnelly, Shania Hayles, Kacie Elson, Deanna Cooper
