The git checkout master of the answer by Woland isn’t needed. Instead use the “Checkout to specific local branch” in the “Additional Behaviors” section to set the “Branch name” to master.
The git commit -am "blah" is still needed.
Now you can use the “Git Publisher” under “Post-build Actions” to push the changes. Be sure to specify the “Branches” to push (“Branch to push” = master, “Target remote name” = origin).
“Merge Results” isn’t needed.