XDA member TGA_Gunnman has just posted up his one-click root application for the Samsung Captivate. This is good news and makes your life and gaining root a whole lot easier. I know I always get a few people wishing to find these things for Mac and Linux, Â so let me put this out in advance. This a Windows based application. Sorry guys.
Here are the steps which are directly from xda-developers website.
Root:
1. Make sure your phone is on USB Debug mode: (MENU > Settings > Applications > Developement > USB debugging = Checked )
2. For Windows x86 and x64 systems make sure that the Samsung Drivers are installed. You can get them here.
3. Extract the contents of GalaxyS_Captivate_One-Click_Root.rar to a folder.
4. Connect your phone to your PC via the USB cable and launch the file “AT&T Captivate One-Click Root.exe” file.
5. Click the “One-Click Root” button. This will launch a command prompt follow the instructions in the CMD window.
You can download the application here.
The only current PC requirement is .NET Framework 4.0, which if you don’t have, you can get here: Dot Net Framework version 4
This has only been tested on the stock Android 2.1 Eclair. So once the Android 2.2 Froyo update starts to roll out, I’d highly recommend NOT using this method until it is updated with Android 2.2 support.
Source: xda-developers