Mark Stoneman hit 78 not out as Durham reached 163-3 in reply to Somerset's 458 by stumps on day two at Taunton.
Opening batsman Stoneman received good support from Scott Borthwick, who contributed 56 to a second-wicket stand of 102 before his 64-ball effort was ended by spinner George Dockrell (1-33).
Somerset seamer Jamie Overton (2-33) had removed Stoneman's opening partner Keaton Jennings for 15 and added the wicket of Will Smith (7) with the final ball of the day.
Nick Compton had earlier been dismissed for 166 as Somerset missed out on maximum batting bonus points despite turning their overnight first-innings position of 336-5 into a final total of 458.
They only managed to reach 379-5 by the end of the 110th over, paying the price for a lack of acceleration as England Test opener Compton, in particular, declined to change tempo from the first day.
Compton, on 139 at the start of the day, was among young off-spinner Ryan Buckley's haul of 3-128, while Peter Trego (42), Dockrell (31) and Overton (24) all added lower-order runs.