Well, so I was able to get it working.
After researching I found out there are a few options to go by, mainly creating your own local prod build and uploading it and using CD/CI. I went with the latter. It took quite some time, but now it's all set I don't have to worry about it anymore...
I based myself in this tutorial http://tattoocoder.com/angular2-azure-codeship-angularcli/ by Shane Boyer but since link-only answers are discouraged I'm going to write it here.
Here's how I did it:
- Create a branch release on GitHub (I use this one to publish)
Create a free account on CodeShip and import the GitHub repo

On Configure Project
select I want to create my custom commands
and use this code:

nvm install 4.1
npm install angular-cli
npm install
- Then this one under Test pipeline:

ng serve &
ng e2e
ng build -prod
- Click
Save and go to dashboard
- Now go to your Azure Portal (https://portal.azure.com/) and open/create your web app
- Click on
Deployment Options > Choose Source > Local Git Repository

- Then click on
Deployment Credentials
and insert the user/password you prefer

- Click on
Overview
and copy your Git clone url

- Go to
Project settings > Environment variables
and add AZURE_REPO_URL
with the value being the git clone url you copied with the user/password (https://username:password@site.scm.az(...).git
):

- After, click on
Deployment
on the left navigation menu
- Choose the branch you want to deploy from (in my case it was release) and click Save, then click on
Custom Script

- Then customize and add this script:
git config --global user.email "email@provider.com"
git config --global user.name "Your name"
git clone $AZURE_REPO_URL repofolder
cd repofolder
rm -rf *
cp -rf ~/clone/dist/* .
git add -A
git commit --all --author "$CI_COMMITTER_NAME <$CI_COMMITTER_EMAIL>" --message "$CI_MESSAGE ($CI_BUILD_URL)"
git push origin master
That was it. Now every time you push to GitHub CodeShip will build your code and every time you PR to release it will build AND publish to azure.
Thank https://stackoverflow.com/users/595213/shayne-boyer for this.