Somerset required only 34.2 overs at Taunton to dismiss the visitors, who were chasing an improbable 418.
Leach took three for 14 while fellow England international Bess claimed two for 12.
Essex are 15 points behind Somerset after winning by an innings and 121 runs against Nottinghamshire.
Simon Harmer ended the match at Trent Bridge with figures of four for 35 after the hosts had resumed at 100 for five.
Ravi Ashwin scored 35, but bottom-of-the-table Nottinghamshire were bowled out for 183.
Surrey lost seven wickets for just 23 runs as Yorkshire secured a thrilling victory at Scarborough.
The visitors, who had been set 318 to win, were cruising at 157 for two before their dramatic collapse left them clinging on for a draw.
Jamie Smith and Gareth Batty kept Yorkshire at bay for nearly 10 overs before Duanne Olivier made the winning breakthrough by removing the latter with only 10 balls left.
Dominic Sibley scored a career-best 244 as Warwickshire drew against Kent at Canterbury.
Sibley spent almost 11 hours at the crease as Warwickshire declared at 574 for seven, 11 fewer than Kent made in their first innings.
In Division Two, Marnus Labuschagne starred for Glamorgan as the Welsh county extended their lead at the top of the table with a draw against Worcestershire.
Labuschagne became the first batsman for Glamorgan since 2005 to score a century in both innings, hitting 100 in his second outing as his side declared on 246 for 5.
Worcestershire were set a target of 326 and they were 143 for one when they shook hands, with Callum Ferguson and Daryl Mitchell unbeaten on 70 and 64.
Lancashire are three points adrift of Glamorgan after Cameron Bancroft hit an unbeaten 92 for Durham to deny the Red Rose victory.
Durham's Nathan Rimmington quickly finished off Lancashire's tail – with England's James Anderson batting with a runner after his injury problem on Tuesday.
Lancashire were dismissed for 247 but the victory target of 304 left Durham with most of the final day to survive if they were to avoid defeat.
Bancroft then starred as Durham reached 194 for six after 75 overs to prevent Lancashire from taking the victory.
Meanwhile, captain Dawid Malan missed out on the first double ton of his career, as his Middlesex side drew against Derbyshire.