Number 11 batsman Barnes first hit the eight runs at the bottom of the order to get his side up to 181 all out in the face of 4-47 from evergreen Thushara Managei.
Weybridge skipper Joel Pope’s men had little answer to Barnes’ exploits with the ball in hand as he claimed 5-44 alongside team-mate Matthew Cawood’s 4-53 to halt the response eight runs shot of the victory target.
The result kept Sutton’s faint hopes of chasing down league leaders Reigate Priory alive and condemnd Weybridge to a season of mid-table mediocrity.
Malden Wanderers head to Esher for the clash of the day in Division One tomorrow, with both teams in desperate need of a win.
Giles Puckle’s Wanderers are only 10 points above second-bottom Esher after a six wicket defeat at Cranleigh for whom ex-Surrey and Weybridge man Seren Waters (4-47) and Brandon McLean (4-43) starred.
Esher had Peter Scott (56) and Jonathan Barrett (52) to thank as they secured a winning draw against Old Whitgiftians at New Road.
Dan Taylor (5-68) and Nicholas Winder (4-54) dismissed the visitors for 214, but Esher fell three runs short of the winning score as they were held to 212-9 with the last pair standing.
Third-placed East Molesey’s four wicket defeat at league leaders Normandy leaves them 19 points short of the Division One promotion places.
Fixtures: Weybridge v Guildford, Esher v Malden Wanderers, East Molesey v Cheam