The soldier known as Hekmatullah fled after shooting dead the Australian soldiers and wounding two others on a base in 2012 and was sentenced to death in 2013.
Bagram district chief Darwaish Raufi said Sunday that the surrender handed the one-time American base over to the insurgents.
Taliban negotiators headed to the presidential palace Sunday to discuss the transfer, said an Afghan official who spoke on condition of anonymity for fear of reprisals.
Once supplied to see off Taliban, abandoned US weaponry now fuelling militant push against Afghan forces
The Taliban's social media is awash with videos of Taliban fighters seizing weapons caches -- the majority supplied by Western powers.
Dozens of civilians have already died in the intense battle for Lashkar Gah, a city of 200,000 people that would be the Taliban's biggest prize since they launched a nationwide offensive in May.