Deputising for the injured Sophie Luff, stand-in Somerset captain Niamh Holland won the toss, inserted the opposition and watched her bowlers reduce the visitors to 30-2 in 4.1 overs.
Lauren Winfield-Hill plundered three boundaries in moving effortlessly to 19 from 14 balls, only to be lbw to an Olivia Barnes delivery that pinned her in the crease.
Three balls later, fellow opener Grace Scrivens followed the former England batter back to the pavilion after pulling a shortish ball from Erin Vukusic to mid-wicket for six.
Charged with the task of repairing the damage, Griffith and Grewcock advanced the score to 40-2 at the end of the powerplay and 66-2 at the halfway stage, despite finding boundaries hard to come by.
The 50 partnership occupied 45 balls and Griffith, having raised 31 from 29 balls, was beginning to accelerate when she attempted to pull Mollie Robbins to the deep mid-wicket boundary and succeeded only in finding Alex Griffiths.
That was the cue for Grewcock to take centre stage, the England A batter opening her shoulders to harvest three successive fours in an over from Barnes as the fielding side came under real pressure for the first time.
Grewcock was eyeing a half-century when she gave the charge to Wellington’s leg spin and holed out to Vukusic at long-on.
Essex were 115-4 in the 16th over with work still to do as Jo Gardner joined Maddie Penna in the middle. They staged a useful stand of 40 from 27 balls, Australian Penna scoring 21 at a run a ball and the forthright Gardner 22 from 12 deliveries, to hoist Essex to a competitive total.
Somerset required the reassurance of a good start and Bex Odgers and Holland provided exactly that, adding 41 in five overs before the former blotted her copybook and top-edged to short third man for 20 off the bowling of Eva Gray.
Holland had also scored 20 when she surrendered her wicket in similar fashion next over, skying a catch behind off Esmae MacGregor as the home side slipped to 41-2.
Buoyed by her half century against Hampshire last time out, Fran Wilson hit the ground running as Somerset passed 50 in the seventh, while debutant Ruby Davis calmed any nerves by straight-hitting Gray for four as the third wicket partnership began to profit.
They added 46 in 36 balls and were threatening to take the contest by the scruff of the neck when Davis, on 19, attempted to reverse sweep MacGregor and was adjudged lbw with the score 87-3 in the 12th.
Wellington drove Grewcock down the ground and pulled Smale for four to serve notice of her intentions, but Abtaha Maqsood and MacGregor kept things tight to push the asking rate above eight.
But Somerset’s fourth wicket pair took 15 off the 15th, bowled by Smale, at which point they required 35 from five.
Victory looked to be a formality, only for Wilson to dance down the wicket and lose off stump to MacGregor with 24 needed off 22 balls.
Wellington then hit Scrivens to long-on in the final over, but Griffiths and Chloe Skelton saw the job through.
Report by ECB Reporters’ Network, supported by Rothesay
