The Australian smacked 78 from 44 deliveries, while Dan Mousely's unbeaten 72 from 36 balls lifted the Bears to an imposing 5-203 on Sunday.
Webster's countryman D'Arcy Short, who cracked an unbeaten in Gloucestershire's 57-run victory over Surrey the previous day, contributed 25 in reply for the Central Group table toppers, but his side ultimately fell short.
Adam Lyth struck a stunning unbeaten 131 to power Yorkshire to a 41-run win over Leicestershire at Headingley.
The opener's second Blast ton lifted Yorkshire to 4-207, with Hasan Ali and former Australia star Andrew Tye both taking four wickets apiece to limit the Foxes to 9-166 in reply.
Central and West table-toppers Northamptonshire Steelbacks suffered their first defeat of the season after they were dealt a 105-run defeat by reigning champions Somerset.
Thomas Rew was the top scorer for the hosts with 57 from 30 deliveries, and Josh Thomas ended with four for six as Northants were bowled out for 118, former Australia Test opener Nathan McSweeney adding 14.
South group leaders Hampshire secured another victory, beating Kent by 19 runs. The hosts finished on 3-190 thanks to half-centuries from captain James Vince and Joe Weatherley, then held Kent to 171 all out, with Australian Tom Rogers scoring 13.
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