Both of the Toffees’ new recruits, Kathrine Kuhl and Rikke Madsen, were eligible and featured from the bench in the second half, but it was fellow Dane Olesen who set the ball rolling for the Blues with a smart finish from the centre of the 18-yard-box after being found by Hanna Bennison.
Snoeijs doubled her side’s advantage at Bescot Stadium on 80 minutes with a measured effort before the Dutch star converted from the spot seven minutes later after being fouled by Villa’s Anna Patten.
Though the first 45 ended goalless both sides worked their opposing ‘keepers, with Villa’s Rachel Daly and Everton’s Clare Wheeler having chances in the opening ten minutes.
Referee Amy Fearns was busy with her whistle in a first half that saw the official blow up for a number of niggly fouls, something which prevented a real flow in the game.
But it was visitors Everton who were coping better and Villa stopper Anna Leat had to be sharp to deny both Emma Bissell and Snoeijs in quick succession.
Arguably the Blues’ best chance of the first half fell to leading scorer Megan Finnigan. But the skipper fired over from close range after the hosts failed to clear a corner.
Into the second half and it was the Villains who fashioned the first chance through Daly. The England forward’s header was a strong one but Courtney Brosnan was equal to it.
Two minutes later and it was Everton who were to be denied, this time by the frame of the goal. Bennison unleashed a drive from distance that cannoned off the post with Leat beaten.
Olesen wasn’t to be denied on 58 minutes, though. This time Bennison turned provider and produced a cut-back from the by-line that the Dane converted with a unstoppable finish to give her team a deserved lead.
The goal provoked a response from the Villains and Brian Sorensen rang the changes. Lucy Hope replaced Elise Stenevik, while Kuhl and Madsen made their debuts at the expense of goalscorer Olesen, and Bissell, respectively.
And the lively Madsen looked to make an immediate impact on proceedings but saw her shot blocked by the Villa rearguard.
But they could do little to stop Snoeijs making it two on 80 minutes. The forward initially fed Kuhl but saw the ball rather fortuitously return into her path and made no mistake with a composed finish.
And the former PSV and Bordeaux star had the final say on proceedings when she slotted her spot kick down the centre of Peat’s goal after being felled by Patten to round off a convincing victory underpinned by a performance that Sorensen will want as a benchmark for the rest of the season.
The Blues will have to wait until 30 January to discover who they will face in the fifth round, but have a WSL date with Arsenal on Saturday, 20 January, at Meadow Park, with Kick off at 14.00 (GMT).