Burly all-rounder Ian Blackwell struck his second Championship century of the season to help Somerset to a lead of 69 at 385-7 with a day left at Hampshire.
After large chunks of the first two days were lost to rain, Somerset needed quick runs as they resumed on 41-0.
Opener Marcus Trescothick hit 11 fours and a six in his 65 before he swept spinner Liam Dawson to mid-wicket.
But Blackwell remained on 104 from 174 balls, hitting a six and reaching his 20th career ton with his ninth four.
The bonus points secured by the visitors, promoted from Division Two last season, took them temporarily to the top of the Division One table.
Left-arm seamer James Tomlinson, who was awarded his county cap at the start of this match, picked up Arul Suppiah and then Zander De Bruyn for a fifth ball duck when the score was 55.
James HIldreth hit five fours in his 50 but was run-out by a direct hit from Dawson at cover.
Spinner Imran Tahir removed Wes Durston, lbw trying to sweep, and had Craig Kieswetter caught at bat and pad.
Peter Trego shared 88 with Blackwell before he was Tahir's third victim but Blackwell, who hit the spinner over long-off for six, added an unbroken eighth-wicket stand of 46 with Alfonso Thomas.
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