Put into bat, Leicestershire resumed day two at 132 for four but lost Dinesh Mongia to the seventh ball of the day. Mongia scored 41 from 77 balls that included four hits to the fence.
Down the order, stumper Paul Nixon hit a crucial 72 off 150 balls with nine boundaries in it to swell the score to 254.
Seasoned seamer Andy Caddick took three wickets, while his pace colleague Charl Willoughby scalped four for Somerset.
Somerset's was not a sound reply as Stuart Broad came up with a three-wicket haul in his seven-over burst and it was Arul Shappiah's unbeaten 63 and the opener's 98-run partnership with skipper Ian Blackwell (49) which brought some respectability to the score which read 138 for five.
In another Division Two match, Zaheer Khan's Worcestershire seems heading for a draw with Glamorgan at New Road.
In reply to Glamorgan's first innings total of 239, during which Zaheer claimed four wickets for 65, Worcestershire were all out for 252.
In their second essay, Glamorgan were 90 for two with Zaheer claiming the wicket of David Hemp (10) who perished caught behind.
Zaheer's second innings figures read 8-2-36-1.