Lancashire’s Grace Potts bowled superbly to take a career-best 5-20 as The Blaze were restricted to 135-9 despite Beaumont’s 59 off 40 balls, which included eight fours.
But the feat was trumped by Australian all-rounder Knott who turned the game with a triple-wicket maiden in the seventh over of Thunder’s reply, including the key dismissal of her compatriot Meg Lanning (21).
Threlkeld kept the visitors in the hunt with 49 from just 36 balls, but she fell to Grace Ballinger (2-26) in the 17th over attempting a sixth boundary.
Georgia Elwiss (3-12) then cleaned up the tail to give Blaze a fifth win from six and move them a point clear of Surrey.
A fourth win in five games for Somerset was underpinned by spinners Liv Barnes (2-20) and Chloe Skelton (2-26) as Warwickshire were restricted to 123-8.
Meg Austin and Mary Taylor top-scored with 31 apiece, but the visitors, who had won the toss, struggled to build partnerships.
Bex Odgers got the reply off to a flyer with 25 off 16 balls, including five boundaries, before Holland took over, helping Somerset rack up 52-2 in the powerplay.
The in-form all-rounder struck five boundaries in a 31-ball knock of 42 and dominated a third-wicket stand of 54 with Anika Learoyd (20) as the hosts condemned Warwickshire to a sixth successive defeat.