England captain Edwards led the way with 151 off 142 balls and was well supported by international team-mate Beaumont, who made 107 from 129 deliveries.
The final eight wickets added only 30 thanks in the main to Erin Bermingham's 4-28 as Sussex lost the top-of-the-table contest by 82 runs.
Another England star, Amy Jones, hit a century in an emphatic 311-run victory for Warwickshire against Essex.
Jones struck 124 from 80 balls in an imposing 360 for six before Essex crumbled to 49 all out, Warwickshire's opening bowlers Laura Crofts and Georgia Hennessy each taking three wickets.
Nottinghamshire moved up to second in the table after they clung on for a two-run win against Middlesex.
Danielle Wyatt was the star for Notts as her 4-41 helped bowl Middlesex out for 192 with a ball to spare. Amy Gauvrit ended the contest when she bowled Danni Warren.
Opener Lauren Winfield lead the way with 128 from 130 balls as Yorkshire claimed a first success by 46 runs against Berkshire.
Aimee Simms' 3-42 helped restrict Berkshire to 207 for nine in reply.
In Division Two, Fran Wilson's 45 led Somerset to a five-wicket win over Wales and Devon number three Rosalie Fairbairn hit 96 not out in a losing cause as Lancashire won by the same margin.
Charlotte Whyle struck an unbeaten 108 to take Staffordshire to victory by seven wickets against Worcestershire, while Durham defended just 133 to beat Cheshire by seven runs on Saturday.
In Division Three, Hertfordshire cruised to a 10-wicket win over Gloucestershire, Derbyshire beat Hampshire by four wickets and Leicestershire & Rutland enjoyed a four-wicket success against Northamptonshire.
Northumberland edged to a two-wicket triumph over Cumbria in the only match in Division Four - North & East.