Is it charged to 100%? Does it show as charging in the taskbar?
See https://support.microsoft.com/en-us/help/402353... and the section on removing the battery driver and reinstalling it and checking for updates.
If it is charged to 100% or is near 100% and shows as charging but still turns off when unplugged after doing the above, if a two button reset doesn't fix the issue (it probably won't), you need to exchange your Surface.
Step 1: Press and hold the power button on your Surface for 30 seconds and then release it.
Step 2: Press and hold the volume-up button and the power button at the same time for at least 15 seconds and then release both.
The screen may flash the Surface logo, but continue holding the buttons down for at least 15 seconds.
Step 3: After you release the buttons, wait 10 seconds.
Step 4: Press and release the power button to turn your Surface back on. You should see the Surface logo.
If the above still does not fix and if you are in the US, I strongly suggest you go to a physical retail Microsoft Store and speak to a store manager and consider swapping for the newer Surface Pro 2017 if at all possible. https://www.microsoft.com/en-us/store/locations... - outside the US, contact support https://support.microsoft.com/en-us/help/13948/.... If you phone support, they will most likely ask you to reimage from a downloadable recovery image https://support.microsoft.com/en-us/surfacereco... to rule out software issues. But you would need a hardware exchange if the trouble shooting above failed. Going to a store (no matter where you purchased) is the best option.
Sorry you are having this problem. Hope this helps. If it was of assistance, please DO mark as Answer.