Josh Coulson put the visitors ahead early in the game but Tom Derbyshire equalised for Grasshoppers just after the interval.
However, Brodie Doherty grabbed the all-important winner for RI to give them three points and move them up to fifth place.
Tom Kilvington scored the decisive strike to give Haxby United’s survival hopes a welcome boost as they defeated bottom club Church Fenton 1-0.
Paul Smith, Chris Wiseman and Gary Simpson scored for Magnets with Russ Carr and Rich Carroll replying for the hosts.
Aviva edged the bottom of the table clash with Stamford Bridge by the odd goal in five.
Jimmy Dunwoody, Ryan Davidson and Neil Davidson were on target for Aviva, while the replies for the visitors were shared by Anthony Downing and Kevin Smith.
In reserve ‘C’, basement outfit Strensall gained a much needed three points after edging a seven-goal tussle against visiting St Clement’s.
Charlie Coupland and Liam Longdon pocketed two goals apiece in the 4-3 win with the St Clement’s booty going to Stuart Kehoe, Dylan Henderson and Joseph Hamilton.
Tockwith prised a surprise point after they drew 1-1 with leaders Thorpe United.
The only reserve ‘D’ game saw F1 emerge triumphant by 3-1 over Thorpe.
F1 fell behind early on but countered with three late goals from Dan Cooper, Will Neviani and Ben Calverley (penalty).
Old Malton St Mary’s are through to the North Riding County Cup Final after they defeated Whinney Banks 4-3 on penalties following a 2-2 draw in normal time.