Curtis Larner was on target twice as Bedale reduced the deficit on Rawdon to just five points, with two games in hand.
Goals from Ross Hodgson, Jonathan Rayner and substitute Stuart Adams completed the rout, which saw Bedale race into a three-goal cushion at the break.
The defeat saw Leyburn fall to the foot of the table on goal difference, below fellow North Yorkshire outfit Kirkby Malzeard.
Hampsthwaite United continued their strong start as they overcame Wetherby Athletic A 2-0 on the Stray, despite the referee bringing the match to a halt on 75 minutes.
Kirk Deighton Rangers leapfrogged opponents Harlow Hill with a 2-0 victory away from home.
Rangers’ Reserves ended a run of seven league games without a win with their first three points of the season over Ripon Red Arrows.
The second-string pulled alongside Harlow Hill Reserves at the foot of the table, with a tally of five points from seven matches.
Bramham and Pannal Sports played out a 1-1 draw, while Dalton Athletic and Boroughbridge A also finished level by the same score.
Braces for James Ford and Alasdair Bowles took Beckwithshaw Saints Reserves away from the relegation zone with a 4-3 win over Pateley Bridge.
Two goals from Scott Lobsinger helped Bardsey Reserves overcome Addingham 4-2 to move up to fourth.
Callum Johnson’s hat-trick paved the way for FC Harrogate to dish out a 7-2 win over Kirkby Malzeard Reserves.
Two late goals from Chris Simpson took Helperby United to a 3-2 win over Pool A, with Dan Marston also on the scoresheet.