The win earns Surrey, who are joint top of the Championship and leaders of the Sunday League, a home semi-final against Essex on 13 August.
Stewart admitted afterwards: "I'm desperate for us to win something, as a player and as a captain. When we won the NatWest in 1982 I wasn't part of the side. We're challenging for three competitions and I'll be happy to win any one of them."
Surrey resumed on 126 for 4, with 25 overs in which to knock off the required 100 runs to pass Somerset's 225, made yesterday before bad light ended proceedings early.
Butcher, the man of the match, added 39 more runs to his overnight 52 as he and Hollioake took Surrey to within 22 runs of victory. At that point Butcher was stumped off a Keith Parsons wide after spending more than three hours at the crease.
Hollioake saw them through, though, finishing 45 not out as they reached 226 for 5 with more than seven of their allotted 60 overs remaining.
Lancashire yesterday welcomed another Roses battle with Yorkshire after the arch-rivals were paired together for next month's NatWest Trophy semi- finals. Mike Watkinson's side, who beat Yorkshire by a wicket at the same stage in the Benson and Hedges Cup last month, will again have home advantage following the draw.
NATWEST TROPHY Semi-final Draw: Lancashire v Yorkshire (Old Trafford); Surrey v Essex (The Oval). Ties to be played on 13 August.Reuse content