Paul McBean’s men edged a game lacking in both goalscoring chances and quality with two second-half goals.
The first came in controversial circumstances as referee John Chalk adjugded Rich Garton’s effort to have crossed the line, much to the frustration of the Falsgrave players.
Duchess wrapped up the points after 85 minutes when Adam North found the back of the net.
McBean singled out defender Craig Sherwin as his side’s star man.
Another side in good form are Cayton Reserves, who beat Snainton 4-0.
New gaffer Damon Cook has rejuvenated the Division Two side since taking over, and they picked up another three points against struggling Snainton.
Michael Dennis opened the scoring with a quickly-taken free-kick from just inside the Snainton half that was helped over the keeper by the windy conditions.
Jack Pinder rifled into the top corner to hand Cayton a 2-0 lead at the break before goals from Josh Sands and Michael Szalabski after the break rounded off the scoring.
Cook singled out 16-year-old Ellis Howley after a superb performance in midfield.
Filey Town Reserves came away from their trip to Edgehill Reserves with the spoils after a 2-1 win.
Goals from Town’s Joe Gage and Edgehill’s Tom Scales had the scores level at 1-1 until Dave Hooper fired in the winner for Lee Cappleman’s men.
Ayton Reserves battled hard but eventually succombed to a 2-1 home defeat to Seamer Reserves.
Much-improved Ayton led 1-0 at half-time through Paul Oldroyd’s strike, but second half goals from Matty Dawson and Ryan Jackson handed Seamer the win.
Keeper Shaun Dolan was in fine form for relegation-threatened Ayton.
West Pier Reserves’ home clash against Newlands Reserves was called off.
Division Three promotion-hopefuls FC Rosette gained revenge for their defeat to Sherburn last weekend with a 3-2 victory this time out.
Gary Tyson’s tap-in handed Sherburn the lead, but the game was turned on its head by two long-range stunners from Al Wray just before the break.
Liam Scott’s bullet header made it 2-2, but Rosette weren’t to be denied and Chris Thompson was first to react after a fumble from the Sherburn keeper to tap in and seal the win for his side.
Martin Russell and Dean Byerley stood out for Rosette.
Hunmanby United netted a 4-0 win from their trip to Sleights Reserves.
Cameron Dobson netted a brace for Dave Wedge’s men, while goals from Reggie Smith and Dean Coates handed the away side the win.
Nick Ringrose had to be at his best in goal for United to keep the scoreline at 4-0.
Eastway Sports switched their game against Scalby to their Eastfield base and maintained their title push with a 3-0 win.
Scalby edged the first half, but a couple of mistakes from defender Karl Hansell handed Eastway two glorious chances to take the lead, and they did just that.
Another goal late on rounded things off for the home side.
Nick David and John Bateman stood out for Scalby.
Westover Wasps Reserves clash against Filey Town 3rds was switched to Filey after all games at Oliver’s Mount were called off.
The sides couldn’t be separated despite Wasps leading 2-0 with just 10 minutes left on the clock.
Goals from Aaron Howard and Grant Hayden handed Town a share of the spoils.
Fishburn Park’s home game against Commercial was called off due to high winds after 28 minutes.
The home side led 2-0 when the referee called a halt to proceedings.