After winning only two of their opening six games, Ed Joyce’s men needed a victory – but instead slipped to a fifth defeat in seven as they faltered chasing a target of 164 to win.
England wicketkeeper-batsman Craig Kieswetter starred in the Somerset innings of 163 for four with a knock of 70 but no-one in the Sussex reply could match his big hitting.
They ended all out for 129, with Yasir Arafat top-scoring with 36 not out. It leaves the Sharks seventh in the table halfway through their campaign.
Sussex are at Arundel for the week, with a four-day county championship game against Yorkshire following on Monday.