Regular wickets plagued Durham during the early stages of their innings, despite a half-century from skipper Hollie Armitage (58).
However, in the latter stages, Bess Heath (51) and Mia Rogers (56) smashed quick-fire fifties to take Durham close to 300 having been 164-5 at one point.
Somerset’s chase failed to get going, mainly down to the brilliant bowling of Levick, with Niamh Holland’s 67 in vain.
At the Kia Oval, Surrey skipper Kira Chathli returned to the side with her highest score of this year’s One-Day Cup to secure victory against Lancashire.
Chathli, who sat out Tuesday’s win over Essex, struck an unbeaten 88 from 101 balls, while Australian all-rounder Maitlan Brown signed off her stint at Surrey with 62 from 74 as the pair added 134 from 154 for the fifth wicket.
Earlier, Brown’s efforts of 3-46 with the ball – in tandem with leg-spinner Dani Gregory’s 3-42 – helped restrict the visitors to 246-9, built around Eve Jones’ third half-century of the tournament.
The Lancashire opener top-scored with 69, sharing a partnership of 70 from 86 with Grace Johnson, but it was not enough to stave off the reigning and inaugural champions’ third consecutive defeat.
At Scarborough, fourth-placed Warwickshire claimed their fourth win of the season as they handed Yorkshire a fifth defeat in eight games.
An outstanding performance from the hosts’ all-rounder Jess Jonassen – who scored a third century of the season with 110 from 102 balls and took three wickets – was not enough for Yorkshire as Chloe Brewer’s 56 steered the visitors home.
Queenslander Jonassen put on 89 for the second wicket with Georgie Boyce and 79 for the fifth wicket with Ines Blackwell but Yorkshire were all out for 238 with four balls remaining.
Warwickshire’s reply got off to a dreadful start when Georgia Redmayne was run out off the first ball of the innings but Amu Surenkumar (45) and Brewer helped them ease to victory at 239-7 with 16 balls left.