Simone Magill opened the scoring with a shot across goal from Lucy Graham's looping pass, before the former Robin turned from creator to scorer and tapped home Nicoline Sorensen's cross in the 19th minute to double their lead.
Then Graham found herself on the scoresheet once more, slamming home a penalty on the hour mark after Magill was tripped in the box.
Debutant substitute Valerie Gauvin made sure to add insult to injury late on with a low finish, from a second Graham assist, that helped push Everton to the second in the Women's Super League, with their hosts on the opposite end of the ladder.
Circumstances were vastly different from the norm. In these two's first game since lockdown began, they emerged from different tunnels, one after the other, with no handshakes allowed.
The visitors nearly got off to a flyer as Moe Wold's cross was headed behind at the far post by Sorensen but, following Aimee Palmer's wayward effort at the other end, did take an early lead as a looping pass found Magill through on goal and she made no mistake with the finish.
Palmer went close again, this time with a curling free-kick from the left-corner of the penalty area, before the Toffees scored a second with a swift counter. Sorensen picked out former-Robin Graham, who coolly side-footed home from 12-yards.
Everton were good value for their lead, Graham nearly secured a third goal contribution for the day, but her point-blank header was well held by City's Sophie Baggaley.
As the half-time whistle sounded, the visitors went into the break with a deserved lead, though the Robins had shown glimmers of quality. Their key to unlocking the visitors' rigid backline was Palmer. For Everton, Graham was untouchable.
It was instead up to Carla Humphrey in midfield to notch up the first of City's chances at a comeback, her effort from the box was tame and easily gathered, though.
Jemma Purfield's shot was a tad stronger a few moments later. The debutant smashed towards goal from 20-yards and Alexandra Maciver was forced to stick out a leg.
But despite their hard work, Faye Bryson's trip on Magill in the box meant the visitors earned a penalty, which the impressive Graham converted, owing to the aid of the crossbar, to seemingly put the game out of sight.
The heavens then began to open to make matters worse for the struggling Robins, as another former City player, Poppy Pattinson was introduced into the action to shore up an already solid Everton back-line.
They then extended their lead even further in the opening seconds of stoppage time, as substitute Gauvin slotted below Baggaley from Graham's pass. Ebony Salmon's similar effort up the other end was less successful.
And that was that, as Everton walked into the Twerton Park dressing rooms the deserved winners of their opening WSL contest.
The stats that matter
This is Everton's third consecutive victory over the Robins - and without conceding a goal as well. 2-0 and 5-0 victories last season carry on City's barren run against the Toffees.
What does it mean?
Only one set of fixtures have been completed, but this was some statement of intent from the Toffees. They looked quick, skillful, creative and were clinical in front of goal.
Graham could be key this season. Four goals and an assist last term were added to in stylish fashion, with an assist and two well-taken strikes either side of half-time.
One game is hard to judge sides on but after a poor offensive record last season, this performance looked all too familiar for the Robins. They couldn't make their chances count and were carved apart in defence. The loss of Pattinson to their opponents could be crippling.
Meanwhile, Everton host Spurs, who finished a place behind the Toffees last term.
Player of the Match - Lucy Graham
The former Robins midfielder came back to bite her old club, as her four goal contributions were the difference between the two sides.
A classy lobbed pass was dispatched early on before a coolly taken side-footed effort of her own 12 minutes later.
Then her powerful penalty on the hour mark put the game out of sight.
Finally a threaded through-pass into Gauvin gave the former Montpellier forward her first Everton goal.