The 1.02 patch has been released on the PlayStation Network. CVG's test team say the file is 39MB in size.
In a post on the Rockstar news wire, the developer said the Xbox 360 version would be available "as soon as possible today".
"Thank you everyone for continuing to bear with us since yesterday's initial roll out of Grand Theft Auto Online - we're aware that a variety of technical difficulties have made it difficult to get access and that the experience has been a bit less than smooth for many of you who jumped in straight away," the company said.
It then listed four technical issues that the 1.02 update is hoped will resolve.
For PlayStation 3 users, a PlayStation Network server issue was detected that was limiting the amount of possible concurrent users. This was preventing most players from accessing GTA Online altogether and as of this morning, that problem was resolved.
There is an issue that is preventing some players from getting through the first tutorial mission across both platforms. Many players have been able to get through this with repeated retries, and we are working on unblocking this for everyone as soon as possible.
This morning, we experienced some instability across the GTA Online services which is affecting even people who have progressed beyond the tutorial mission. We are identifying these issues now and are working to restore stability for users as soon as possible.
For the time being and until we have been able to get everybody access to GTA Online and things are running smoothly, we have disabled the option of purchasable GTA$ cash packs. Players can however keep on earning GTA$ by pulling off Jobs and other profitable gameplay activities rather than purchasing cash packs.
Update 2: CVG's test team say online games are now more stable on PS3, though the wait in the lobbies can last as long as ten minutes. These post-patch release issues could be improved over time.
Update 3: As of 12:44pm UK time, the Xbox 360 update is still unavailable.