Touchscreens on smartphones that have this problem gradually become unresponsive.
The fault was highlighted by gadget sites which said it was caused by a manufacturing issue that meant some screen controller chips became loose.
Apple was criticised by one expert who said it was the phonemaker's responsibility to fix it.
Apple is charging $149 in the USA for the service that will only be available on phones that are "in working order".
In a statement on its website, Apple said it had "determined" that displays on some iPhone 6 Plus handsets flickered or became unresponsive if the device was dropped several times on a hard surface and was then subjected to "further stress".
An iPhone 6 Plus suffering this problem, which worked and did not have a cracked or broken screen, was eligible for the repair programme, it said.
The service is available worldwide and costs Â£146.44 in the UK.
Apple said that customers who had already paid to get their phone cured of "touch disease" should get in touch to be repaid for the amount they spent beyond the programme fee.
The repair programme will run for five years beyond the initial date on which the iPhone 6 Plus went on sale, said Apple.
Stuart Miles, founder of gadget news site Pocket-lint, said: "I think if it's a known defect that isn't caused by the user then it should be the responsibility of Apple to fix the phone free of charge regardless of the age."
The manufacturing problem with the iPhone 6 Plus was first highlighted by gadget site iFixit which said it had seen an "influx of faulty iPhones".
In a blogpost about the problem posted in August, iFixit said there were "pages and pages" of messages from aggrieved iPhone 6 Plus owners on the Apple support site complaining about the fault.
An investigation by iFixit and specialist repair shops found the cause of the fault was the placement of the touchscreen controller chip. Several rough knocks could dislodge this chip making phones unresponsive, it said.
Apple has started offering a service to fix a fault on iPhone 6 Plus phones known as "touch disease".