Nottinghamshire were set a target of of 255 at the Cooper Associates County Ground.
One of the two overseas players for the club, Ashwin took three for 93 in the first innings.
Nottinghamshire were bowled out for 241 in their first innings after Somerset's 326 all out, conceding a lead of 85. Besides the 32-year-old Ashwin, who had a match haul of eight for 123 in 64.3 overs, left-arm spinner LA Patterson-White picked up five for 73.
This is Ashwin's second match for his county team after taking three for 162 in Nottinghamshire's innings defeat to defending champion Essex.
Meanwhile, Hampshire's overseas player and Indian batsman Ajinkya Rahane failed against Warwickshire at Hampshire Bowl in Southampton.
Batting at No. 3, Rahane made four in the first innings before being caught off the bowling of seamer Olly Stone. In Hampshire's second innings, Rahane was trapped in front of the wicket for three by Oliver Hannon-Dalby.