Here are some steps:
Get a custom TWRP Recovery
1. Download and unzip Odin 3.09 (Use this, MUST)
2. Get ready your device and boot into Download Mode. (Press Volume Down, Power and Home button together)
3. Press Volume UP after you in Download Mode.
4. Connect your device to computer now by using USB cable.
4. Open Odin with Auto-reboot and F.Reset time enabled. (Auto-reboot Uncheck/Check here doesn't matter if using updated Odin 3.09 here)
5. Select AP and choose TWRP recovery. You can download from here. (Thanks to @gr8nole)
6. Click 'Start' afterwards.
7. Done. Now you should have custom recovery working.
Note: Don't use CWM recovery as CWM is not working properly.
1. Boot into TWRP recovery by pressing Volume Up, Power and Home button together.
2. Flash supersu.
3. Restart system. You now have root working. SuperSu apps should installed on your device.
Feel free to ask any question here. But I expect you know root will get warranty void and I'm not responsible for any damage you get, e.g. Soft Brick. Thanks.
[HowTo] Root Galaxy Tab 3 Lite - 7.0 (SM-T111), Tool/Utility for the Samsung Galaxy Tab 3 7.0
Last Updated 2014-11-07