Beckwithshaw remain the closest rivals to the league leaders Otley as they beat an out-of -form North Leeds side by seven wickets.
North Leeds were bowled out for 157 with Yasar Ali collecting figures of 4-22.
In reply Beckwithshaw strolled to victory after 32 overs, Alexis Twigg was the pick of the batsmen finishing on 67.
Elsewhere in Division One, top-of-the-table Otley kept their pursuit of another league title on track as they won a low-scoring game against Rawdon.
Otley batted first but could manage a total of only 163 from their 50 overs with Josh Atkinson top-scoring on 57 not out.
In reply, however, Rawdon lasted only 28 overs as they fell 56 runs short.
Chris Thompson (4-32) and John Brough (3-7) did the main damage with the ball for the league leaders.
Horsforth scored a more convincing victory as they beat lowly Steeton by 170-runs.
Horsforth batted first and an impressive 161 from just 129 balls from Oli Hardacker helped the home side post a total of 331-4. In reply, second-bottom Steeton lasted 39 overs before being bowled out for 161.
Pool cooled their relegation fears as they condemned bottom-of-the-table Kirkstall Educational to a 99-run loss.
Jake Zjilinszky top-scored for Pool with 162 not out in a result which leaves Kirkstall 36 points adrift of safety.
Collingham have slipped to 10th place as they were leapfrogged by Ilkley after a seven wicket defeat.
Burley have also kept up their outside hopes of the title as they edged past Bilton by two wickets.
All teams in Division Two were also in action last weekend, and it looks like it’s a two-horse race for the title between Colton Institute and Guiseley.
Colton Institute currently lead the way after they beat Skipton by 119-runs.
Captain Daniel Gautrey led the field with a score of 105.
Guiseley have kept up the pressure by securing a close-fought victory over Addingham.
Having set a total of 283, Guiseley hung on to win by 14 runs – despite the best efforts of Will Atkins, who scored 110 runs.
The result means Guiseley now trail Colton by nine points with eight league games remaining.Elsewhere in Division Two, Thackley won a crucial fixture in the race to avoid relegation. They beat second-bottom Illingworth by five wickets. There were also victories for Ben Rhydding, Calverley St Wilfrids and Adel.