The agreement limits TfL's contribution to the construction of the bridge to £10m rather than the original £30m contribution. Any money spent in excess of the £10m will be treated as a loan and will be paid back.
The decision means that the negotiations for the bridge's south landing site in Lambeth will resume.
Lambeth Council leader Lib Peck had suspended negotiations around the lease because of growing concerns about the £30m TfL had contributed to the scheme.
Cllr Peck said: "I'm pleased that Londoners are getting a better financial deal particularly at a time of austerity when all public sector organisations are being forced to make deep cuts to services.
"We've been in tough negotiations with the Garden Bridge Trust and Transport for London and I'm pleased we've successfully agreed a deal that will cut London taxpayers' contribution towards the Garden Bridge by two thirds."
In a statement on the Lambeth Labour website Peck said £10m was "still too much in my view", but added: "Taking into account the amount of money that has already been spent, this was the best possible deal we could have reached."
The trust anticipates work beginning on site early in 2016. Chairman Lord Mervyn Davies said the trust is delighted to be moving forward.
"We have been hugely successful in our efforts to raise funds from the private sector, with £85m pledged to date, and we have agreed that any of the committed funds from TfL spent over the £10m will be treated as a loan. We are delighted the Garden Bridge can now progress and are grateful for all the support we've had."
More sponsorship announcements from private investors are in the pipeline, the trust has told HortWeek.
The National Audit Office has separately confirmed it is looking at the rationale for the Treasury investing a separate £30m of public funding in the project.