Derbyshire chased down 365 for victory.
Somerset and Yorkshire are also closing in on opening-round victories.
Former Yorkshire and England all-rounder Bresnan joined Warwickshire this summer.
County cricket returned to England on Saturday.
The new-look Bob Willis Trophy has replaced the traditional Championship campaign in the wake of the Covid-19 pandemic.
Clarke wants Surrey, who were due to welcome 2,500 fans into Kia Oval this weekend, to honour Bob Willis by winning the competition named after him.
The campaign was meant to start in April.
Surrey and Middlesex meet again and the Bears take on Northamptonshire in the new first-class competition.