Reigate & Banstead Borough Council has dumped its green-waste bags to reduce landfill waste and cut down on manual work for collection crews.
Residents asking for the 240-litre bins have to pay for the fortnightly service, which aims to make green-waste storage and collection easier, safer and more convenient.
Councillor Julian Ellacott, said: "In just a few short weeks we have had over 5,000 households sign up for the service. "We can provide a more efficient service, offer less manual handling for our refuse crews and reduce the number of plastic bags going to landfill."
The waste bins take grass cuttings, hedge trimmings, weeds, prunings, dead plants, twigs and small branches less than 10cm in diameter.
Others switching from bags to bins include Epson & Ewell Borough Council, which launched its brown tubs in late 2006 to improve collection. The London Borough of Sutton, meanwhile, is piloting use of jute collection bags, with residents paying £35 per 75-litre bag.